design recommendations for concrete tunnel liningsvolume ii: summary of research and proposed...
TRANSCRIPT
u.s. Department of Transportation
Urban Mass Transportation Administration
UMT A-MA-06-0100-83-3 DOT -TSC-UMT A-83-16
PB84-158047
Design Recommendations for Concrete Tunnel Linings Volume II: Summary of Research and Proposed Recommendations
S.L. Paul A.J. Hendron E.J. Cording G.E. Sgouros P.K. Saha
University of Illinois at Urbana-Champaign Department of Civil Engineering Urbana IL 61801
November 1983 Final Report
This document is available to the public through the National Technical Information Service, Springfield, Virginia 22161.
UMTA Technical Assistance Program REPROOUCED BY NATIONAL TECHNICAL INFORMATION SERVICE
US DEPARTMENT OF COMMERCE ... SPRINGFIElD, VA. 22161
NOTICE
This document is disseminated under the !Sponsorship of the Department of Transportation in the interest of information exchange. The United States Government assumes no 1 i abil ity for its contents or use thereof.
NOTICE
The United States Government does not endorse 'products or manufacturers. Trade or manufacturers' names appear herein solely because they are considered essential to the object of this report.
, .'
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Technical ~eport Documentation Page
1. Report No. 2. Government Accession No.
UMTA-MA-06-0l00-83-3
4. T oil. and Subti tl.
Design Recommendations for Concrete Tunnel Linings. Volume II: Summary of Research and Proposed Recommendations~
7. Author's) S. L. Paul, A. J. Hendron, E. J. Cording; G. E. Sgouros~ and P. K. Saha
9. Perlorming OrganIzatIon Name and Addre ..
3. RecIpIent's Catalog ,Illc-.
5. Report Dote
November 1983 6. PerformIng OrganIzatIon Code
DTS-77 8. Performing Organization Report No.
DOT-TSC-UMTA-83-l6
10. Work Unit No. (TRAIS)
University of Illinois at Urbana-Champaign* Department of Civil Engineering
MA-06-0l00(UM476/R4642)
Urbana, Illinois 61801
12. Sponsoring Agency Name and Addre ..
U.S. Department of Transportation Urban Mass Transportation Administration 400 Seventh Street, S.W. Washington, D. C. 20590
15. Supplementory Notes U.S. Department of Transportation
11. Cont'oct or Grant No.
DOT-TSC-1S04 13. Type 01 R.port and P.riod Cover.d
Final Report April 1978-December
14. Sponsoring Agency Code
URT-10
*Under contract to: Research and Special Programs Administration Transportatinn Systems Center Cambridge, Massachusetts 02142
16. Abstract
1982
The overall objective of this research effort was to formulate recommendations for the structural design of final concrete linings for tunnels and underground stations for mass transit use that are based on ultimate strength concepts of concrete behavior and that take full advantage of interaction between the linings and ground. The report describing this work is in two volumes. Volume I contains a detailed description of the test arrangements, results, and numerical studies.
This report, Volume II, presents design recommendations for concrete tunnel linings for transportation tunnels. The recommendations developed as a result of in-depth analysis and model testing of the behavior of concrete tunnel linings. The research addressed problem areas in current rlesign practice, and the authors point out that the ~esu1ts have provided insight into the areas of uncertainty that have led designers to gross overconservatism in tunnel lining design.
The recommended procedures take into account ultimate strength behavior of reinforce and unreinforced concrete linings and provide sufficient latitude for designers to exercise judgement gained through experience and allow the flexibility required by site-specific conditions. Details of the suggested approach are based on procedures that have been accepted for years in the design of above-ground structures, with appropriate modifications to capitalize on the benefits of ground/structure interaction.
17. Key Words Concrete Tunnel Linings; Ground/Struc-ture Interaction; Tunnel Design; Tunnel Linings; Ultimate Strength Design
18. Distribution Statement
Available to the Public through the National Technical Information Service, Springfield, Virginia 22161.
19. Security Clossi!. (01 this report) 20. Security Classi!. (of this page) 21. No. 01 Pages 22. Price
Unclassified Unclassified 176
Form DOT F 1700.7 (8-72) Reproduction of completed poge outhori zed
\ \
PREFACE
The studies described in this report were performed by the staff of the
Department of Civil Engineering of the University of Illinois at Urbana
Champaign, Urbana, Illinois during the period April 1978 to April 1983. The
work was sponsored by the U.s. Department of Transportation, Urban Mass
Transportation Administration under contract with the Transportation Systems
Center, Cambridge, Massachusetts and performed under the technical direction
of Mr. Gerald Saulnier and Ms. Beth Madnick. Their helpful suggestions,
advice and patience are gratefully acknowledged.
This organization and the technical monitors organized a review of the
research and recommendations in June 1982 that was attended by many members
of the tunnel design community. This meeting was extremely helpful in further
assessment of the recommendations and evaluating reaction by designers. For
these efforts, we are grateful to the organizers and to the many engineers who
took the time to review the report and attend the meetings.
During the project, a review of tunnel design practice was conducted by
Dr. George Sgouros by interviewing members of several design firms and contrac
tors. We wish to acknowledge the assistance of the tunnel design and con
struction community and thank those who participated for their time and
assistance.
iii
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TABLE OF CONTENTS
SUMMARY
CHAPTER 1 INTRODUCTION
1.1 OVERVIEW
l.2 GROUND BEHAVIOR
1.2.1 Loading on Concrete Linings
l.2.2 Ground Loads: General Considerations
1.2.3 Behavior of Linings in Soil
l.2.4 Behavior of Linings in Rock
1.3 PROCEDURES FOR ANALYZING LINING-MEDIUM INTERACTION
1.3.1 Introduction .....
1.3.2 Closed Form Solutions
1.3.3 Beam-Spring Model
1.3.4 Nonlinear Analyses
CHAPTER 2 EVALUATION OF EXISTING DESIGN PRACTICE
2.1 PURPOSE AND METHODOLOGY
2.2 LOADING
2.2.1 Tunnels in Rock
2.2.2 Tunnels in Soft Ground
2.3 ANALYSIS
v
xv
1
1
5
5
7
8
14
19
19
22
26
3]
35
35
39
39
41
42
2.3.1 Tunnels in Rock 42
2.3.2 Tunnels in Soft Ground 43
2.4 DESIGN CRITERIA. 44
2.5 AREAS OF UNCERTAINTY IN DESIGN 45
CHAPTER:3 SUMMARY OF RESEARCH AND DESIGN IMPLICATIONS 49
3.1 MODEL TESTS 49
3.1.1 Arch Linings in Hard Medium 49
3.1.2 Circles in Soft Medium 59
3.2 ANALYTIC STUDIES 70
3.2.1 Parameter Study of Arches 70
3.2.2 Parameter Studies of Circular Linings in Soft Ground .............. . 85
3.2.3 Parameter Study for Circular Linings in Rock 113
CHAPTER 4 DESIGN RECOMMENDATIONS 129
4.1 SCOPE . 129
4.2 GROUND LOADS AND BOUNDARY CONDITIONS FOR ANALYSIS l31
4.2.1 Soil l32
4.2.2 Rock l33
4.3 LOAD FACTORS l33
4.4 SECTION CAPACITY l35
4.5 MOMENT-THRUST PATH 137
4.6 ANALYSIS OF LINING AND COMPARISON WITH ULTIMATE CAPACITY 140
4.7 EVALUATION OF STRAINS AND CRACKING 145
4.8 APPLICATION OF ACI CODE 148
4.9 SUMMARY 152
BIBLIOGRAPHY \ . 15:3
vi
'.;" -' ..
FIGURE
1.1
1.2
1.3
3.1
3.2
LIST OF FIGURES
LONGITUDINAL DISTRIBUTION OF DISPLACEMENT NEAR THE TUNNEL FACE (FROM RANKEN. GHABOUSSI. HENDRON, 1978)
HOOP LOAD INTERACTION BETWEEN THE GROUND AND LINING AND LOAD-TIME CURVE FOR THE LINING (FROM PECK. 1969)
DEFINITION OF TERMS FOR THE EXCAVATION LOADING SOLUTION FOR INTERACTION OF A THIN LINING AND MEDIUM
TEST SET UP FOR ARCHES
LOAD SHAPES USED IN ARCH TESTS
3.3 EFFECTS OF LOADING SHAPE AND FLEXIBILITY RATIO ON THE
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.11
3.12
3.13
3.14
3.15
3.16
3.17
THRUST RATIO
FIRST CRACKING AS A FUNCTION OF FLEXILIBITY RATIO
TEST SET UP FOR CIRCULAR LININGS
LOAD-DEFLECTION CURVES FOR THE MONOLITHIC LININGS .
CRACKING PATTERN OF CIRCLE-1
CRACKING PATTERN OF CIRCLE-2
CRACKING PATTERN OF CIRCLE-3
EFFECT OF AMOUNT OF REINFORCEMENT ON CROWN CRACK SIZE OF MONOLITHIC LININGS . . .. . . . . . . . .
COMPARISON OF LOAD-DEFLECTION CURVE ON MONOLITHIC AND SEGMENTED MODELS CIRCLE-2 AND CIRCLE-5
CRACKING PATTERN OF CIRCLE-4
CRACKING PATTERN OF CIRCLE-5
FINITE ELEMENT MODEL FOR PARMIETER STUDIES OF ARCHES
MOMENT-THRUST PATHS FOR CRITICAL SECTION FOR VARIOUS LOAD SHAPES . . . . . . . . . . .
TOTAL LOAD VS FLEXIBILITY RATIO FOR ARCHES
THRUST RATIO VS FLEYIBILITY RATIO FOR AFCHES
vii
9
9
25
50
52
55
58
60
63
65
65
65
66
68
69
69
71
73
75
76
FIGURE
3.18
3.19
3.20
3.21
3.22
3.23
THRUST RATIO VS TOTAL ULTIMATE LOAD FOR ARCHES
VARIATION OF RADIUS CHANGE RATIO AT FAILURE WITH FLEXIBILITY RATIO .. . .... ....
EFFECT OF K /K ON MOMENT-THRUST PATHS FOR ARCHES t r
EFFECT OF K /K ON THRUST RATIO FOR ARCHES t r
EFFECT OF REINFORCEMENT ON MOMENT-THRUST PATHS FOR ARCHES ... .... .. ...
EFFECT OF REINFORCEMENT ON THRUST RATIO FOR ARCHES
3.24 EFFECT OF ¢ AND c ON RADIAL PRESSURE AND TANGENTIAL STRESS
3.25 EFFECT OF ¢ AND c ON AXIAL THRUST AND MOMENT
3.26 EFFECT OF SLIPPAGE ON ULTIMATE LOAD . . .
3.27 EFFECT OF WATER PRESSURE ON MOMENT-THRUST PATHS .
3.28 EFFECT OF LOADING CONDITION ON TOTAL ULTIMATE LOAD
3.29 EFFECT OF COEFFICIENT OF EARTH PRESSURE ON TOTAL LOAD .
3.30 PREDICTION OF FAILURE OF LININGS BY THE NONLINEAR ANALYSIS AS A FUNCTION OF THE LINEAR ECCENTRICITY DIVIDED BY THICKNESS. . . . . . . . . . . . . . .
3.31 COMPRESSION STRAIN VARIATION ALONG THE MOMENT-THRUST
77
79
80
81
82
84
88
89
90
91
94
94
96
PATH . . . . . . . . . . . . . . . . . . . . . 98
3.32 COMPARISON OF LOADING CONDITIONS FOR K /K OF 0 M~D 0.125 FOR GRAVITY LOADING IN TERMS OF p7f~r . . . . . 99
3.33 COMPARISON OF LOADING CONDITIONS FOR K /K OF 0.25 AND 0.40 FOR GRAVITY LOADING IN TERMS OF plf,r. . . . . 100
c
3.34 COMPARISON OF GRAVITY LOADING SHAPES FOR K /K OF 0 AND 0.125 t r . . . . . . . . . . . . . . . . . 102
3.35 COMPARISON OF GRAVITY LOADING SHAPES FOR K /K OF 0.25 AND 0.40 t r
103 . . . . . . . . . . . . . . . . 3.36 EFFECT OF K /K
t r FOR THE VARIOUS GRAVITY LOADING SHAPES. . 104
3.37 TENSION STRAINS ALONG THE M-T PATH FOR A REINFORCED LINING . . . . . .. . ............. . 105
viii
FIGURE
3.38 TENSION STRAINS ALONG THE H-T PATH FOR AN UNREINFORCED LINING 105
3.39 EFFECT OF REINFORCEMENT ON MOMENT-THRUST PATHS FOR OVERPRESSURE LOADING ... ,. ... ,. ..... . 107
3.40 EFFECT OF REINFORCEMENT ON TOTAL LOAD AND CRACKING LOAD. 108
3.41 EFFECT OF REINFORCEMENT ON FIRST CRACKING FOR
3.42
3.43
3.44
3.45
3.46
3.47
3.48
3.49
3.50
3.51
OVERPRESSURE LOADING
MOMENT COEFFICIENT VS MODULUS OF LINING FOR AN 8 IN. (200 mm) MONOLITHIC LINING IN DIFFERENT SOIL MEDIA
EQUIVALENT MODULUS OF ELASTICITY OF SEGMENTED LININGS
FINITE ELEMENT MODEL OF CIRCULAR LINING FOR PARAMETRIC STUDIES IN ROCK FOR LOOSENING LOAD . . . . . . . .
LINING CAPACITY AS A FUNCTION OF FLEXIBILITY RATIO
TOTAL LOAD VS FLEXIBILITY RATIO . . . . . .
LINING CAPACITY AS A FUNCTION OF TANGENTIAL TO RADIAL STIFFNESS RATIO (K IK ) FOR R/t = 11.8
t r
EFFECT OF F ON MOMENT-THRUST PATHS AT CROWN .
EFFECT OF F ON MOMENT DISTRIBUTION AT MAXIMUM LOAD
EFFECT OF F ON SHEAR AT MAXIMUM LOAD
EFFECT OF REINFORCEMENT ON MOMENT-THRUST PATHS
4.1 LINEAR AND NONLINEAR M-T PATHS AND PREDICTION OF
4.2
4.3
4.4
FAILURE ......•...
BEHAVIOR OF AN UNREINFORCED SECTION WITH e£ > 0.5t
COMPARISON OF LOADING CONDITIONS AS FLEXIBILITY VARIES IN TERMS OF p/f~ FOR GRAVITY LOADING AND Kt/Kr OF 0.25 ••
COMPARISON OF GRAVITY LOADING SHAPES AS FLEXIBILITY VARIES IN TERMS OF T IT AND FOR K IK OF 0.25
U 0 t r
4.5 PREDICTION OF FAILURE OF LININGS BY THE NONLINEAR ANALYSIS AS A FUNCTION OF THE LINEAR ECCENTRICITY
109
111
112
114
116
118
119
121
123
126
127/128
138
138
141
141
DIVIDED BY THICKNESS . . . • . • . • . • • . . • • 142
4.6 TENSION STRAINS ALONG THE M-T PATH FOR A REINFORCED LINING . . • . • • 4 • . . . . . . . . . , . 146
4.7 TENSION STRAINS ALONG THE M-T PATH FOR AN UNREINFORCED LINING .... 146
ix/x
,-("'
TABLE
2.1
3.1
3.2
3.3
3.4
LIST OF TABLES
LIST OF PARTICIPANTS . . . . . •. .
SUMMARY OF ARCH TESTS IN HARD MEDIUM •
SUMMARY OF ARCH TEST RESULTS . . . .
SUMMARY OF MODEL CIRCULAR LINING TEST RESULTS
EFFECT OF WATER PRESSURE ON VARIOUS PARAMETERS
Preceding page blank xi
Page
37
53
54
62
93
~'- .
LIST OF SYMBOLS
A area of lining/unit length
A area of the lining reinforcement/unit length s
B width of the tunnel opening
b length of lining under consideration in the longitudinal direction
C compressibility ratio for the lining-medium system
C arc length of the arch that is under compressive load from the footing o
to the point where separation of arch and medium occurs (normally
about one-third the lining diameter)
Cl width of the footing of an arch-shaped lining
c cohesion of the medium material represented by the interface element
between the lining and soil
D diameter of a circular tunnel lining
E~ modulus of elasticity of the lining
E in situ modulus of elasticity of the medium m
e~ eccentricity of thrust in a concrete section from a linear analysis
F flexibility ratio for the lining-medium system
f' compressive strength of the concrete c
f yield strength of the lining reinforcement y
G shear modulus
H depth of the tunnel from the surface
HT height of the tunnel opening
I moment of inertia/unit length of lining
~ stiffness of the spring that represents the footing of an arch-
shaped lining
K lateral pressure coefficient in the medium o
K radial spring stiffness representing the medium r
xiii
Kt tangential spring stiffness representing the shear stress between
the lining and medium
k modulus of subgrade reaction
P total load applied to the lining
p = pressure applied to the lining in the gravity loadin~ analysis
Pa air pressure used inside a tunnel
R mean radius of the lining
T thrust at a critical section in the lining due to the applied active a
ground pressure
T ultimate thrust capacity if there is no moment acting at the section o
T ultimate thrust capacity of the lining at a critical section u
t thickness of the lining
u displacement of the medium at the tunnel sides when the lining and p
medium make contact
ut
total displacement of the medium at the tunnel sides if there is no
support provided
y unit weight of the medium
Yw unit weight of water
8 angle sub tended by one beam element in the beam-spring model
E maximum compression strain in the concrete lining c
Et
maximum average tension strain at the tension face of the concrete
lining
v£ Poisson's ratio of the lining
V in situ Poisson's ratio of the medium m
p ratio of area of reinforcement to lining area/unit length of lining
~ = angle of internal friction of the medium material represented by the
interface element between the lining and soil
xiv
. ,
. -; "
SUMMARY
The research described in this report was sponsored by the Urban
Mass Transportation Administration of the U.S. Department of
Transportation, and performed under the technical direction of the
Transportation Systems Center, Cambridge, Massachusetts. The objective
of the work was to provide recommendations for structural design of
final concrete linings for tunnels and underground facilities for mass
transportation use, that include ultimate strength design and
ground-lining interaction concepts, and to improve the understanding of
lining behavior as it interacts with the ground. The need for a set of
guidelines and recommendations evolved from the perception that there
has been little uniformity of approach to lining design that has
sometimes led to over design and poor economy. Also, many designers
have indicated a desire for a reasonable set of flexible guidelines
that can form a norm of standard practice from which designers can
deviate as local conditions vary.
The report describing this work is in two volumes: this one, Volume
II, contains a summary of the research and the design recommendations,
while Volume I contains a more detailed description of the test
arrangements, results, and numerical studies.
In this volume, general recommendations are made for the design of
concrete linings for tunnels in rock and soft ground with a typical
diameter of 20 ft (6 m), and for large openings in rock with spans of
40 to 60 ft (12 to 18 m). A step-by-step method is not given in order
to allow the designer freedom ~n adapting the approach to local
geology, specific requirements of the supports or his own views of
support behavior; however, two analysis approaches are described, one
for rock and one for soft ground, that take into account interaction of
the lining and ground and are adaptable to most geologic settings. The
primary focus ~s on the structural behavior of the lining and the
effect that the ground and the excavation and support process have on
this behavior. Though typical loadings are discussed and their
influence on the lining strength has been investigated, specific
xv
recommendations for loading to be used are not made because they are
dependent on local geology and construction procedures.
The recommendations for design are based on the same general
philosophy as that used for above-ground concrete structures and
described 1n the ACI Building Code Requirements for Reinforced
Concrete, ACI (318-77). That is, the loads that are expected to act on
the lining during the service life of the lining are selected and
multiplied by load factors that depend on the confidence the designer
has in their accuracy. These load factors are intended to make the
lining strength sufficiently high that the service load stresses are
low enough to prevent exceSS1ve creep, cracking, failure under
accidental overload or inaccuracy 1n the analysis procedure. With
these loads, an analysis is performed that accounts for interaction
between the lining and medium and utilizes both the moment and thrust
capacity of the lining. Once moments, thrusts and shears are obtained
from the analysis, they are compared with the corresponding strength of
the lining sections, reduced by a factor that accounts for possible
variation 1n material strength or inaccuracy 1n the strength
computations.
The analyses recommended for use in the design of final linings 1n
rock and 1n soft ground are different because of the way loads are
applied and interaction occurs between the lining and ground in the two
cases. In rock, the final lining is placed in a stable opening and is
initially unstressed; subsequent loading must result from rock that has
loosened or relaxed and rests on the lining, and interaction results
from the lateral pressure between the medium and lining as the lining
deforms. Linings 1n soft ground are loaded by deformation of the
medium and therefore the loading cannot be separated from the
interaction process; also the vertical loading 1S accompanied 1n most
cases by an 1ncrease 1n horizontal pressure in addition to the
horizontal pressure resulting from deformation of the lining and the
vertical loading is changed by the lining deformation.
xvi
.'. ,
" . ,
It is recognized that a m~n~mum constructible concrete lining ~s
adequate ~n many cases, such as circular linings ~n competent rock or
stiff soil, and that an analysis may not be necessary for these cases.
However, if a verification of this idea is needed, the opening is
large, the shape of the opening is not circular or the modulus of the
medium ~s low, then the described design approach is recommended.
Special loading cases, such as squeezing or swelling ground, require
special considerations and are not treated specifically.
The design recommendations are based on studies that can be divided
into three separate categories. These are first a survey of existing
design practice in which 16 design firms and several contractors were
interviewed; the second is a series of laboratory model tests of tunnel
linings in various media; the third ~s development of a computer based
finite element analysis that was used to perform parameter studies to
investigate the major variables affecting the problem.
During the interviews of design firms, it was found that most
designers contend that the final lining of m~n~mum thickness for
convenient construction, varying from 8 to 12 ~n. (200 to 300 mm), ~s
adequate for runn~ng tunnels in rock, though some of them do analyze
them for rock loads and/or water pressure loading. Large openings ~n
rock are normally designed for rock loads that depend on the geology at
the site, though there is considerable variation in the loads selected.
Of n~ne firms who described their analysis for large openings, six of
them used a series of beam elements to represent the lining, applied
loads directly to the lining and represented the medium by springs or
~n one case by continuum elements and in another case considered the
rock to be rigid.
There was greater variation ~n the approaches to soft ground lining
design. Seven firms described the loadings used, of which four used
full or partial overburden depending on the depth for the vertical
load, and applied lateral pressure equal to some portion of the
vertical pressure or determined the design moment by assuming an
ovaling of the lining of a fixed magnitude or applied the thrust
xvii
determined from the overburden at a fixed eccentricity. One firm used
a vertical load of 1.5 to 2.0 tunnel diameters for initial supports and
60 percent of this value for the final lining without applying a
horizontal pressure except for that resulting from ovaling of the
lining during the interaction process. The remaining firms assumed
that a minimum-thickness lining with a predetermined design based on
experience was adequate.
Six firms described their analysis for linings in soft ground of
which three used beam elements to describe the lining and radial
springs for the soil, and of these, two apply a lateral pressure 1n
addition to that resulting from ovaling of the lining while the
remaining one does not, but the vertical load 1S limited in that case.
Thirteen firms described their strength criteria for checking the
sections of the linings after the analysis 1S performed for both
running tunnels in soil and rock and large openings 1n rock. Seven
firms use working stress design methods, four use ultimate strength
design and two use both procedures.
Also from these interviews, it appears that there are several areas
of uncertainty either indicated by the variety of procedures used or
statements of the designers. Among these are the use of reinforcement
to prevent cracking and leakage, treatment of external water pressure,
loadings for linings in soft ground, analysis and design of precast
segmented linings, the degree of conservatism in large openings and
treatment of initial supports in the design of final linings.
Tests were performed on models of arches that were six feet
diameter, embedded in a medium to simulate rock or very firm soil, and
loaded 1n the crown region to represent loosening loads with var10US
shapes. The tests showed that when interlocking was present between
the medium and lining, a triangular load with the peak at the crown
resulted 1n the largest moment 1n the lining or lowest ratio of
ultimate thrust to that at failure with pure axial thrust (T /T ). u 0
When the interlocking was removed the thrust ratio was much smaller.
xviii
Stiffness of the medium was found to be the most important parameter 1n
determining lining strength but even with the stiffest medium tested,
only 62 percent of the axial thrust capacity of the lining could be
utilized 1n resisting load. Cracking of the lining due to flexure
appeared to be of little concern in these stiff media, with the first
crack occurring near or above one-half the ultimate load. When the
medium is sufficiently stiff, it was found that there is enough thrust
to prevent tension 1n the lining unless the loading is concentrated or
the medium is soft. Reinforcement served to distribute cracks more
uniformly when they occurred, and therefore keep the individual cracks
finer.
Parameter studies of arches in stiff media, with a finite element
analysis that would allow nonlinear behavior of the lining, served to
confirm the findings of the model tests and allowed the parameters to
be varied over a much wider range. With this program, the lining
behavior was investigated as medium stiffness, radius to thickness
ratio of the lining and tangential stiffness between the lining and
medium were varied.
Circular lining models of 44 inches diameter, embedded 1n a soft
medium that simulated soil, were tested by applying uniform pressure to
the medium surface. It was found that the medium stiffness is the most
important parameter affecting lining strength, even more important than
it was for arches in the stiffer medium. Cracking occurred at a much
lower load relative to the ultimate because of the larger moment to
thrust ratio, and reinforcement served to distribute the cracks and
keep them finer. Tests on segmented lining showed loads comparable to
those of their monolithic counterparts; the reduced strength of joints
appeared to be compensated by the smaller moment in the linings
resulting from the joints. Also the overall deformations were
comparable because the medium 1S the dominant factor in deformation
rather than the lining.
An analysis of circular linings in soft media, that included
nonlinear lining behavior and interface elements to represent the shear
xix
stress between lining and medium, was performed to expand the range of
parameters studied in the model tests. It was found that the angle of
internal friction of the interface elements had a significant effect on
lining strength. The manner in which load was assumed to reach the
lining was also found to have a significant effect and 1S related to
the construction process. External water pressure applied to the
lining in addition to ground loads was shown to improve lining strength
by increasing the thrust, while the moment is unaffected, resulting in
a smaller moment-thrust ratio. The effect of nonlinear behavior could
be studied with the analysis program, and it was found to provide
considerable additional strength over that predicted by a linear
analysis. The effect of placing joints in the lining as in a precast
segmented system was studied and the reduced stiffness for the
particular joint locations considered was determined for various medium
stiffnesses.
XX
" '. . "
CHAPTER 1
INTRODUCTION
1.1 OVERVIEW
The research described in this report was funded by the Urban Mass
Transportation Administration of the U.S. Department of Transportation
and performed under the technical direction of the Transportation
Systems Center, Cambridge, Massachusetts. The realization by these and
other agenc1es that construction costs for underground transit
facilities are increasing much more rapidly than other construction
costs has led to a search for ways to reduce them. Excavation of the
underground opening and its support, if required, is a significant part
of the total cost, and therefore substantial savings might be realized
by more efficient designs of the support system. In addition, review
of the design of existing linings show that there is little uniformity
of approach, and though they are safe, some are more efficient than
others.
These considerations have led to the research effort described 1n
this report, that has the twofold objective: 1) make recommendations
for the structural design of final concrete linings that encompass
ultimate strength concepts and take full advantage of interaction of
the lining with the surrounding medium, and 2) improve the
understanding of lining-medium interaction and lining behavior near the
failure-load range. The research is described 1n two volumes. This
report, Volume II, contains the design recommendations and a summary of
the research findings. Volume I contains the details of the model
tests and parameter studies.
General recommendations are outlined in Chapter 4 for the design of
final concrete linings for tunnels in rock and soft ground and for
large openings in rock. A step-by-step method is not given in order to
allow the designer freedom in adapting the approach to local geology or
specific requirements of the supports; however, an analysis technique
, .. ,'
~s described that takes into account interaction of the lining and
ground and is adaptable to most geologic settings. The primary focus
is the structural behavior of the lining and the effect that the medium
and the excavation and support process has on this behavior. Though
typical loadings are discussed, and their influence on the lining
strength have been investigated, specific loadings to be used are
dependent on local geology and construction techniques used. To assure
that the ideas suggested for the structural design of linings are
reasonable and do not conflict with other practical design and
construction requirements, many discussions have been held
designers of various types of underground supports.
with
Special loading cases, such as squeezing and swelling ground,
require special consideration and are not treated here specifically;
however, many of the conclusions and analysis procedures developed
would be applicable to these cases with proper modifications of the
loading conditions.
The philosophy of the recommended design approach ~s similar to
that of "Building Code Requirements for Reinforced Concrete" (ACI
318-77). This design method consists of selecting loads that are
actually expected to occur and mUltiplying them by load factors; an
analysis of the system is then performed with the factored loads acting
on the structure; the results of the analysis are compared with the
lining strength that has been modified by capacity reduction factors to
account for uncertainties ~n strength. The safety factor results from
combining the load factors and capacity reduction factors. Specific
recommendations unique to lining design are made for the load factors
to be applied and the analysis method to be used.
The analysis procedure suggested for openings in rock and for
gravity loading in soil is based on representing the lining by a series
of beam elements and the medium by a series of radial and tangential
springs and performing a linear frame analysis. The loads may be
assumed to occur from loosened rock blocks resting on the lining or a
mass of soil above the crown that relaxes and rests on the lining
2
,"., 'r ':', ?',
because of high shear stresses 1n the soil. This analysis is adaptable
to variable geology, shapes, loadings, tangential stress conditions at
the lining-medium interfaces, joints 1n the lining, cracking of the
lining and variable lining cross section; even more important, this
analysis contains the important lining-medium interaction components
that occur in the ground for this type of loading and the construction
procedure normally used.
For some other soil conditons or a soft homogeneous rock, where
loads on the lining result from deformation of the medium, the loading
cannot be separated from the interaction process and there are
interaction components that are active and that are not included 1n the
analysis described above. For example, the actual load on the lining
1S not known because part of it is arched around the lining during the
interaction. Therefore, it is necessary to consider the ground to be a
continuum around the lining in order to obtain a realistic analysis.
If the lining-medium system can be approximated by a uniform circular
lining with linear behavior 1n a homogeneous linear medium, then a
closed form solution 1S available (Ranken, Ghaboussi and Hendron,
1978). In many cases, this solution will suffice. The loading is the
in situ medium stresses at the tunnel location. If a more detailed
analysis is required or if the approximation described above cannot be
made, then it 1S appropriate to represent the medium by continuum
elements 1n a finite element analysis. In a soft medium where
horizontal bedding planes or discontinuities may occur (i.e., stiff
fissured clays) or where the cohesion may be low (i.e., loose sands),
soil may rest directly on E the lining as described above for jointed
rock. If this condition 1S suspected to occur, the lining should be
checked for this loading and a correction procedure 1S provided 1n
Chapter 4 to obtain the results of a nonlinear analysis by performing a
linear analysis.
It is recognized that a minimum constructible lining is adequate in
many cases such as for circular linings with a diameter up to 20 ft
(6.0 m) or perhaps larger in competent rock or stiff soil, and that an
analysis may not be necessary for these cases. However, if a
3
verification of this is required, the opening ~s larger or the medium
~s very soft, then the suggested analyses may be used. Performance of
analyses such as those described above require certain lining and
medium properties that may be difficult to obtain with accuracy or
may be quite variable. Normally, however, reasonable upper and lower
limits can be placed on these properties and the analysis can be used
to predict behavior under normal as well as the most unfavorable
combinations of conditions; it may still be found that a minimum lining
is adequate even under the most unfavorable conditions, and so no
further analysis ~s necessary; if it is found that problems may occur
if these conditions exist, a more careful investigation must be
performed.
The recommendations in Chapter 4 are based on a ser~es of model
tests, parameter studies using a finite element analysis, a series of
interviews with tunnel designers and contractors, and a survey of the
literature. The model tests, described ~n Section 3.1, served to
identify overall lining behavior, modes of failure, and provide
strength and cracking information. They also served to verify the
analysis described in Section 3.2. This analysis represents the lining
with a ser~es of beam elements that may have linear or nonlinear
properties. Geometric nonlinearity is also taken into account in the
solution process. The medium can be represented by continuum elements
or by radial and tangential springs. Special elements may be used
between the lining and medium when the medium ~s represented by
continuum elements to model various conditions of slip at the
lining-medium interface.
Seventeen firms that are involved ~n the design of underground
supports of various types were interviewed ~n order to determine what
procedures are now being used and what these firms feel are the most
urgent problems ~n design. These interviews were also helpful in
determining how recommendations would be most consistent with existing
practice and therefore would be most likely to be adopted for future
designs. The results, of these interviews are summarized in Chapter 2.
4
The remainder of this chapter 1S devoted to a discussion of the
background and thinking that has gone into the recommendations
proposed.
1.2 GROUND BEHAVIOR
1.2.1 Loading on Concrete Linings
Support 1n tunnels 1S used in order to (1) stabilize the tunnel
heading and protect the men and equipment operating in the tunnel, (2)
minimize ground movements that can damage structures and utilities, and
(3) permit the tunnel to perform its intended function over the life of
the project. Traditionally, the first two functions are provided by an
initial support system, whereas the third function 1S provided by a
final lining, usually concrete, installed at some time after the tunnel
has been stabilized with the initial support. Linings which serve as
both initial and final support, such as pre-cast concrete segments,
have been finding increasing use on tunnel projects.
The loading on a final lining and its required capacity 1S very
dependent on when and how it is installed and on the loadings that will
occur after it is installed. Often, a final lining installed after the
tunnel has been stabilized by the initial support will undergo very
little additional loading.
The permanent concrete lining, if installed after ground loads have
equilibrated, will be subjected to the loadings due to its own
installation, such as the pressures applied by contact grouting and
stresses due to thermal effects, and by any subsequent changes 1n
compressed air pressure, ground water pressure, time-dependent soil or
rock creep, nearby excavation or superposition of loads, such as fill.
5
Outline of Loading Conditions
Some of the loading conditions that can affect a final concrete
lining are summarized below:
a) If the concrete lining is placed near the advancing heading, it
will essentially serve as part of the initial support. Loads
can develop due to the tendency of the ground to displace
inward around the tunnel heading either due to elastic
deflection, squeezing or loosening of the ground. The
resulting loads are what the initial support system would be
expected to carry.
b) Stresses generated by shrinkage and temperature changes ~n the
concrete as it sets.
c) Grouting pressures (principally due to contact grouting behind
the lining).
d)
e)
f)
Changes ~n water pressure are likely to occur after the tunnel
lining ~s installed if the drainage system clogs, the
dewatering wells are turned off, or the natural drainage into
the tunnel is reduced by the presence of the lining.
Removal of air pressure after the lining ~s installed.
Additional pressure will be applied to the lining due to
changes in ground stresses if a second tunnel passes the lined
tunnel.
g) Continued creep, squeeze or swell of ground surrounding the
tunnel will apply additional pressure to the lining. These
conditions occur principally ~n shales, clays or other
materials which have significant time-dependent behavior.
h) Loss of lateral support of the lining due to adjacent
excavation.
i) Surface loads applied after the lining is ~n place. Fill
placed over a tunnel in soft compressible soils can cause large
increases in loads on a tunnel lining.
j) Transfer of load from the initial support system to the final
lining, either due to creep or deterioration of the initial
lining.
6
Additional Loads for Precast Segmented Lining
a) Handling stresses prior to installation.
b) Thrust from the shield if segments are used as a reaction for
jacking the shield forward.
c) Pressures due to expansion or grouting of concrete segments to
fill the void between soil or rock and the segment.
d) Nonuniform loads and distortions due to incomplete grouting or
misalignment of segments.
1.2.2 Ground Loads: General Considerations
If a tunnel were constructed 1n such a way that a rigid lining
could be installed with no inward movement, the ground pressures acting
on the lining would be the same as the initial stresses existing 1n the
ground before the tunnel was excavated. If inward ground movements are
permitted, either due to the presence of a non-rigid lining, or due to
delay in placing the lining, then the pressure applied to the tunnel
lining would be reduced below the level of the original in-situ stress.
With sufficient inward movement, which may be large or small
depending on the stiffness of the soil or rock, ground loads on the
lining will reduce to a value that is related to the pressure produced
by the weight of soil or rock immediately around the opening; thus the
value is proportional to the size of the opening and 1S an inverse
function of the strength of the soil or rock medium.
In ground that tends to creep, loads may build up with time, to a
value that 1S a function of the overburden pressure, the restraint
provided by the lining, and the creep charateristics of the material.
Thus, the ground loads that can develop on the tunnel lining can range
widely, from full overburden pressure, a fairly concentrated load
applied by a small rock slab, or even to no load for a self-supporting
rock or a lining installed after all ground movements have taken place.
7
The following sections provide background information on the
loadings that can develop for the various ground and lining conditions.
Both the magnitude of the pressure and its distribution around the
tunnel are of concern, as these affect not only the levels of thrust
but also the bending that develops in the linin~. The thrusts and
moments that develop in the lining will be principally a function of
the initial load distribution and the flexibility ratio, a measure of
the relative stiffness of the lining with respect to the soil or rock
medium.
1.2.3 Behavior of Linings 1n Soil
In soils, the tunnel heading is temporarily supported by a shield
with the initial lining installed in the tail of the shield or the
initial support 1S placed as close to the heading as possible.
Subsequently, one of two courses is normally followed. If the initial
support is concrete segments, it may also be used as the final lining
and no additional lining is added; if the initial support is steel ribs
and timber lagging or steel liner plate, it may only serve to stabilize
the opening until a final concrete lining can be placed after the
mining operation is completed. To understand the forces acting on the
initial and final linings and see how they should be designed, it is
first necessary to understand the construction process and what has
occurred in the ground during this process.
As the tunnel heading progresses, overburden stresses tend to push
the face of the excavation inward, so there 1S both radial and
longitudinal movement directly at the face. If no supports were placed
to restrain the ground, the radial movement would increase with
distance behind the face until a constant deformation is reached two to
three tunnel diameters back. The deformation will also increase with
time due to creep, the amount of increase depending on the type of
soil. In soft clay, the increase with time will be appreciable while
with sandy soils it may be small. The longitudinal distribution of
radial displacement is shown for a hypothetical case in Figure 1.1,
8
, ,;
-o oJ: -at C CD ...J -~
~
" c o ...J
at .5 0::
pR
u
r--- ____________ --
Face I I
Contact of Ground and Lining
Tunnel
-------u
FIGURE 1.1 LONGITUDINAL DISTRIBUTION OF DISPLACEMENT NEAR THE TUNNEL FACE (FROM RANKEN, GHABOUSSI, HENDRON, 1978)
up
" ""-Contact of Ground and LininQ
I
I I I
B
u
pR
Swelling Clays
Non -Swelling Cloys
--------- ""C.. Sandy Soils
During Construction
Average Radial Displacement Time
FIGURE 1.2 HOOP LOAD INTERACTION BETWEEN THE GROUND AND LINING AND LOAD-TIME CURVE FOR THE LINING (FROM PECK, 1969)
9
t
taken from Ranken, Ghaboussi and Hendron (1978). Interaction will
clearly depend on when the lining is placed and how much ground
movement has occurred when contact 1S made between the lining and
ground. That is, if the lining is placed when the displacement u has
occurred in Figure 1.1, the interaction between the soil and initial
lining will depend on the ratio of u /u where u 1S the total
displacement if no support was provided. p t t
The displacement
on
u depends p
the distance behind the face that the lining is placed and the void
between the ground and the lining. A void is usually left between the
ground and initial lining, and an attempt is made to either expand the
lining to eliminate it or to fill it with pea gravel and/or grout soon
after the support is placed. The success of this attempt depends on
the rate of movement of the soil and the care exercised during
construction.
The hypothetical ground reaction curve for ring thrust shown 1n
Figure 1.2 is useful to show how the lining and ground interact, (Peck,
1969). The average ring load, if no radial deformation occurs, will be
approximately the lining radius times the mean of the vertical and
horizontal ground pressures; if the vertical pressure is given by y H
and the horizontal by KYH, then the maximum ring thrust at point A is
1/2 H(l + K ) yR, where R is the outside radius of the unlined opening o
and ~ the unit weight of soil. If the boundary of the opening 1S
allowed to displace inward, the corresponding ring thrust required to
prevent further displacement decreases along curve AB. The shape and
position of this curve depends on the stress-strain-time behavior of
the soil and results from a change in the stress pattern around the
opening that allows some load to arch around it and therefore not be
applied to the lining. If pressure were applied to the outside of the
lining after the deformation u has occurred, the lining would deform
along line CD. At the poi~t E, where the two curves intersect, the
internal pressure on the medium and the external pressure on the lining
are in equilibrium.
As displacement is allowed, the lining pressure reduces to a value
which 1S a function of the weight of the material that must be
10
, ,,;
supported immediately around the opening and 1S thus proportional to
YR, the unit weight of the soil times the radius of the opening. In
frictional materials, the minimum value of load can be expressed as Y R
divided by a frictional coefficient, f(¢). This 1S the gravity load
conditions, sometimes referred to as a "loosening condition" although
it can take place without cracking and separation of blocks of soil
from the surrounding medium. In stiffer soils, the deformations
required to achieve the m1n1mum values are quite small and it is
possible, even when the tunneling procedures are designed to prevent
exceSS1ve ground deformations, to approach or achieve the m1n1mum
values. In frictional materials, such as sand, the minimum ground
pressures will be equivalent to the pressures applied by a height of
soil extending approximately one-half to two diameters above the tunnel
crown.
In soft clays, the deformations required to achieve the m1n1mum
values are large, and pressures acting on the lining will be greater
than these values. In addition, because of the time-dependent behavior
of the clay, the lining pressures will tend to increase with time to
values which are a function of the total overburden pressure. Thus, in
many soft clays, final pressures are assumed to be close to the total
overburden pressure.
An important aspect of tunnel construction 1S ground water control,
which may be accomplished by preconstruction dewatering (pumping
through wells installed at the surface) or by pressurizing the tunnel
face with compressed air or slurry-face machine systems. Differences
1n ground water control can produce significant variations 1n
distribution of ground stresses and in the loads applied to the lining,
particularly linings constructed 1n stages. If the excavation is made
under air pressure, the relation between ring load and displacement for
the soil would be displaced downward in Figure 1.2 by the air pressure
times the radius, p R; if the lining and soil make contact when the a
displacement is u , equilibrium would be reached at the point E', but
the load would Pincrease back to E when the air pressure is removed.
Water pressure applied subsequent to construction of the lining would
11
ra~se the curve for the ground thrust by the water pressure times the
radius of the opening, p R and result in equilibrium at a point on CD w
above E.
Passage of time may result in further increase in the ring loads as
shown ~n the right side of Figure 1.2 where the amount of ~ncrease
depends on the type of soil. The ring thrust may approach that for
full overburden in soft plastic clays and may increase very little ~n a
sandy soil.
Though the diagram shown ~n Figure 1.2 would be difficult to
construct for a particular case and does not include the moment arising
from nonuniform loading, it shows the important influence that the
construction process has on the lining load because of its effect on
u, which is made up of the deformation that occurs prior to
l~ning-ground contact and includes the gap between lining and ground.
It also shows that the average overburden pressure ~s a reasonable
upper limit for determining r~ng thrust in the lining even when the
effect of creep is included and that the thrust will almost always be
less than overburden pressure times the radius. The deformation that
occurs toward the lining in a soft soil before it contacts the lining
is usually large in comparison with the deformation of the lining after
contact is made.
In reality, linings are neither perfectly flexible or rigid. The
distortions, moments and thrusts developed in such linings can be
determined by considering the relative stiffness of the lining with
respect to the soil, expressed by the Flexibility Ratio "F". Both
elastic continuum analyses and beam-spring structural models used to
approximate the lining and soil are available and have been commonly
used by designers to evaluate the effect of the relative lining
stiffness on the distortions and moments in the lining for var~ous
loading configurations. Such analyses have proven very useful but have
major limitations when used to evaluate the ultimate capacity of
concrete linings in conditions where the eccentricity of the thrust is
great enough to produce tension ~n the concrete linings. In such
12
cases, concrete linings, particularly when unreinforced, have
capacities well in excess of those determined from the linear analyses.
The results of the research program, consisting of tests of concrete
lining models and analyses that include the nonlinear behavior of the
concrete, have provided information that can be used in assessing the
actual capacity of such linings.
The loading on an initial lining will depend on the amount of
ground movement taking place before contact is made between the lining
and the ground and on the pressures that develop during grouting or
expansion of the lining. Since the deformation and loading depend so
much on the construction process, it can only be estimated or based on
calculations that provide a reasonable upper limit for design. Some
initial supports, such as steel ribs and lagging or steel liner plate,
are very flexible and can deflect sufficiently to resist the ground
loads principally 1n thrust. They continue to deform as the soil
pressure changes due to advance of the heading, changes in the soil
stresses with time, or adjacent construction such as another tunnel.
Segmented linings that may constitute both the initial and final
support may be somewhat less flexible and may suffer loss of strength
at the joints if exceSS1ve rotations occur there; however, lining
deformation before contact with the soil can be controlled by careful
grouting and use of horizontal tie rods. Greater care is likely to be
exercised 1n the installation of initial linings that will also serve
as the final lining.
In some soils, time-dependent relaxation may continue for an
extended period and additional pressure may occur after the final
lining is placed. This additional load 1S resisted by both the
temporary and final linings as a composite structure. Furthermore,
many designers assume that the initial supports will deteriorate so the
soil pressure that was originally resisted by them (or some portion of
it) would be transferred to the final lining. If this philosophy is
adopted, it is the most significant source of loads on the final lining
and in most cases will control its design. In this case, the load
assumed to be resisted by the temporary lining is critical in the final
13
lining design. The initial support 1S usually very flexible and
therefore deforms so that the external pressure is nearly uniform and
it is this nearly uniform pressure that would then act on the final
lining.
1.2.4 Behavior of Linings 1n Rock
Rock excavation 1S
techniques, by road headers
performed primarily by
(boom mounted cutters)
drill and blast
1n some softer
rocks, or by tunnel boring machines where long, circular tunnels are to
be excavated. When drill and blast techniques are used, an irregular
surface on the interior of the opening is usually created while a
fairly smooth surface 1S created with mechanical excavation.
Vibrations from the drill and blast operation are likely to loosen the
remaining rock more than the tunnel boring machine.
Many tunnels are likely to be excavated and supported temporarily,
with a concrete lining placed after excavation is completed; however,
rock tunnels are also supported with permanent linings installed close
to the face. For example, the large openings for the subway chambers
in rock on the Washington Metro were supported initially by rock bolts
and steel ribs shotcreted in place, with the steel ribs and shotcrete
becoming part of the final lining when additional shotcrete was added
to encase the ribs. Another example is the concrete segments used for
both initial and final support in TBM excavated tunnels.
Loosening Ground
Rock may support itself around an opening if it contains few joints
or joints that are discontinuous and irregular, so that a structural
lining may not be necessary. The geologic settings of interest, then,
are those in which sufficient jointing, bedding planes, faults, or
weathering exists so that rock blocks and wedges will be unstable
unless supported.
14
Most rock masses are stiff enough that very little deformation of
the tunnel is required to relieve the original overburden pressures.
Even when efforts are made to install support as soon as possible
behind the heading, enough displacement takes place to relieve most of
the overburden pressures that could act on the lining. In such
conditions, the only loads that must be accommodated by the support
system are those due to the weight of the immediate blocks of rock
surrounding the opening that would tend to displace into the open1ng,
less any shearing resistance developed along the boundaries of the rock
blocks. These gravity loads, often termed loosening loads, are
proportional to the unit weight of the rock times some dimension which
1S a function of the width of the opening or the size of the rock
wedges, and the configuration and strength of the rock discontinuities
surrounding the opening.
If excessive displacements are allowed, the rock around the opening
will loosen, the interlocks and peak strength along joint surfaces will
be lost, and the ultimate lining loads will increase.
One of the earliest methods for estimating rock loads is found in
Tunneling with Steel Supports (Proctor & White, 1945). In this volume,
Terzaghi presented a tunnel ground classification for rock using terms,
such as "intact," "stratified," "massive, moderately jointed," and
"blocky and seamy," that can be considered loosening ground for steel
rib supports. The design rock load recommended for these ground
conditions are given in terms of a height of rock that is proportional
to the size of the opening. The height of rock ranged from 0 to 0.25 B
for "intact" rock, to (B + H ) for massive, moderately jointed rock to T
(0.35 to 1.1) (B + H ) for very "blocky and seamy" rock where B is the T
width of the tunnel and HT is the height. The same rock loads are not
necessarily applicable to other types of linings, because the amount of
ioosening developed with steel ribs and the degree of conservatism in
the other assumptions used in designing ribs may differ from those used
with other types of lining.
15
'.
Cording and Mahar (1978) also evaluate rock loads on the basis of
the sLze of blocks or wedges that are separated by discontinuities
They recommend examining the local geology and around the opening.
selecting typical
the lining based on
rock
the
blocks and wedges that may loosen and bear on
orientation and character of the actual
discontinuities expected at the site. The size of the critical rock
wedges assumed to require support will depend on the orientation of the
discontinuities and the shear strength along the joints. Large, deep
wedges have the potential for failing if the discontinuities bounding
the wedges are planar and sheared. The same shaped wedge could not
fail if it were bounded by irregular or discontinuous joints. Such
joints, however, could allow separation of a thin slab or small rock
wedge located near the tunnel perimeter.
Methods for estimating loads based on rock mass characteristics,
such as RQD, weathering, characteristics of joints, bedding and faults,
and ground water conditions have also been proposed (Barton, et al.,
1974, Wickham, et al., 1974, Bieniawski, 1974).
Tangential shear stress between the lining and rock influences the
lining behavior a great deal and is difficult to evaluate, but in most
cases, a cast Ln place lining will be capable of developing significant
shear along the rock-lining boundary. If the lining is cast directly
against the irregular surface resulting from the drill and blast
technique, it seems reasonable to assume that there would be little
relative tangential deformation and no slip, although incomplete
grouting and the presence of voids could reduce the effective shear
stress between the lining and rock. Even with a smooth TBM bored
perimeter, significant tangential shear stresses will develop. The
assumption of full slip and no frictional resistance between lining and
rock allows large deformation of the lining and therefore large moments
occur, but this condition is unrealistic Ln most cases. By whatever
means the frictional resistance and tangential stiffness along the
contact is taken into account, it is reasonable to assume that they act
not only adjacent to but also Ln the region where the active rock load
is applied.
16
Loads
depending
applied
on the
by loosening rock can
geology of the rock mass.
have different shapes,
Highly fractured layered
strata may loosen and apply load uniformly over the crown region or
triangular wedges may ravel onto the lining and give maximum load at
the crown. Larger blocks or systems of blocks may rema1n relatively
rigid as they rest on the lining; deflection at the crown would then
reduce the load at that point and allow the block to rest primarily on
each side of the crown, resulting 1n a pressure pattern that is minimum
at the crown and increases laterally in both directions. An oblique
system of joints in the rock may cause the load to be concentrated at
one side of the crown region or to be applied on the side of the
lining. If the geologic conditions are known well enough to draw the
possible patterns of joints that may exists at the site on the tunnel
cross-sections, it will be possible to estimate the shape and magnitude
of loadings that are likely to occur.
Methods are available for modeling the development of gravity loads
by simulating the medium with blocks separated by joints that have
nonlinear behavior, and these methods are most desirable when they are
available. However, this type of analysis 1S normally too complex for
the usual application and the programs are not generally available.
Therefore, it 1S reasonable to select a simple procedure that models
the final lining with the loads from rock blocks applied directly to
the lining. The full weight of rock blocks and wedges reduced by
friction and/or interlocking forces will generally be conservative; if
sufficient information 1S available to predict how the load may be
reduced by prompt support with rock bolts, then an advantage may be
taken of these additonal load reductions as well.
Though the initial supports have stabilized the opening by the time
the final lining 1S placed, many designers feel that the initial
support should not be considered as part of the final lining because
there is little control over its placement and/or it may deteriorate
with time. An exception, in some cases, is the inclusion of that part
of the steel ribs that are embedded in the final lining concrete or
17
when steel ribs are used for initial support and become part of the
final lining by adding shotcrete.
This approach seems reasonable in V1ew of the protection that the
concrete offers the steel. Calculations show that a final lining of
m1n1mum thickness is adequate to support the usual loosening rock loads
applied to a circular or near circular tunnel with a diameter of up to
20 ft (6 m) provided adequate account is taken of radial and tangential
passive resistance. Therefore it is usually only for larger openings,
for noncircular shapes, or for heavy squeezing ground that use of the
temporary supports 1n the final lining need be considered. Straight
side walls in the lining are normally used only when the rock 1S quite
competent so that lateral rock loads are not likely to be large. In
this case, water pressure 1S then the primary loading on the walls, if
drainage is not provided.
Squeezing Ground
Squeezing ground 1n rock is often associated with weathered zones
and fault zones in which significant amounts of clay minerals are
present. Squeezing can also take place in weaker rock materials, such
as shale and tuff, which often contain clay minerals as well. In
squeezing ground, the overburden pressures can be relieved only by
relatively large inward displacements, and loads may continue to build
after the lining is installed. The rate of buildup is dependent on the
creep parameters of the material and on lining stiffness. If movements
are prevented, loads will be high. If movements are allowed, the loads
that the lining will carry will decrease, although loads and
distortions will increase if uncontrolled movements are allowed. In
such cases, bending effects can be severe, particularly where there 1S
a great variation 1n rock properties around the perimeter of the
tunnel, such as rock blocks separated by seams and zones of soft soils
and where the support is not in continuous contact with the ground.
Squeezing pressures can be estimated from previous experience and
from time-dependent analyses. They will be some function of the
18
overburden pressure as well as the creep properties of the material and
the amount of displacement allowed to relieve ground pressure. For
deep tunnels (greater than 500 ft deep) passing through creep-sensitive
materials, such as weak tuff or clayey fault gouge, squeezing pressures
as high as 30 to 50 percent of the overburden pressure have been
measured. Smaller values are observed where controlled ground
displacement is allowed.
In many cases, the permanent lining is not installed until the rate
of inward movements has become small. Thus, the only pressure the
lining will sustain will be due to any tendency for small additional
creep or an increase in water pressure on the lining. Practice in some
30-ft wide openings in squeezing ground in Europe is to stabilize the
tunnel with shotcrete, bolts and light steel ribs and then install a
nominal, 10 in. (250 rom) concrete lining after placing a waterproofing
membrane after most of the ground movements have stopped.
1.3 PROCEDURES FOR ANALYZING LINING-MEDIUM INTERACTION
1.3.1 Introduction
Both linear and nonlinear analysis of the behavior of concrete
linings 1n soil or rock are discussed. A major part of the study was
devoted to development of a nonlinear analysis of the ultimate behavior
of circular and arch-shaped concrete linings 1n soil and rock.
Nonlinear analysis of the post-peak behavior of the concrete material
1S required 1n order to evaluate the ultimate capacity of a concrete
tunnel lining, particularly when bending takes place. The analysis
results have been compared with the model test results, and parameter
studies have been performed in order to extend the range of lining
flexibilities and loading conditions.
Linear analyses are also presented and discussed 1n this section,
principally because they are readily available to designers and have
19
been used 1n the past to evaluate lining behavior. The linear analyses
permit a reasonable and conservative estimate of the ultimate thrust
capacity to be obtained, if eccentricities are small enough to result
1n thrusts and moments 1n the compression region of the moment-thrust
interaction diagram (above the balance point). However, linear
analyses may result 1n overdesign and use of excessive amounts of
reinforcement if they are used to evaluate ultimate thrust capacity 1n
cases where eccentricities are high enough to fall well below the
balance point. Procedures have been developed and are presented 1n
Chapter 4 that permit the designer to obtain the nonlinear thrust
capacity once he has performed a linear analysis. A relationship has
been developed between the linear elastic eccentricity obtained from a
linear analysis and the ultimate thrust determined from the nonlinear
analysis.
For the purpose of discussing the analysis techniques used for soil
tunnels, it 1S helpful to first describe the three types of loading
considered. If a region of the medium containing a lined opening 1S
isolated and a pressure 1S applied to the upper surface, it is
considered to be an "overpressure" loading. The lining was placed 1n
the medium when it was unstressed and the lining and medium interact as
the surface pressure 1S applied. Lateral stress 1n the medium 1S
normally handled by applying a lateral pressure to the medium. This
loading represents the vertical stress 1n the soil at the tunnel level
due to the overburden for a deep tunnel.
In reality, the lining is never placed in an unstressed medium, but
the excavation 1S made in the medium with the overburden stresses and
corresponding deformations present. The lining is then placed 1n the
opening after this initial deformation has occurred so that it fits
perfectly and before any additional deformation occurs due to
excavation of the tunnel. This is called the "excavation" loading and
is different from the overpressure loading 1n that the lining 1S not
subjected to the medium deformation that occurs due to the 1n situ
stress before the opening is excavated. The moment and thrust in the
lining resulting from the excavation loading are smaller than those
20
\ \
,.
from the overpressure loading. However, the excavation loading still
ignores the deformation in the medium due to the excavation before the
lining is placed and the void between the lining and medium in soft
ground tunnels discussed in Section 1.2.3. In terms of the deformation
equivalent to
= 0 and the
due to excavation in Figures 1.1 and 1.2, this case 1S
placing the lining ahead of the excavation so that u p
lining fits perfectly in the opening.
In rock and in some types of soil, the weight of a mass of material
above the opening can rest directly on the lining, resulting in a
loading equal to the weight of some depth of material reduced by the
shearing stresses at the edge of the mass. This will be referred to as
a "loosening" load when it occurs in rock and a "gravity" load when it
occurs in soil because the mechanism is different in the two materials.
Once contact is made between the lining and medium, interaction
begins because the lining reduces the deformation of the medium and the
medium reduces the deformation of the lining. For the purpose of
comparing the analysis techniques, it is convenient to separate this
interaction into the following four components in order to see how each
technique handles them.
1) When a load is applied to the crown region of the lining, the
vertical diameter shortens, causing the horizontal diameter to
lengthen and thus push the springline region into the medium.
The lateral medium pressure on the lining thus increases and
results in a stabilizing effect on the lining because it tends
to resist the ovalling. This is the interaction due to overall
deformation.
2) When the lining ovals, the crown region deforms so that the
crown deflects more than the adjacent regions near the crown
causing a change in shape of the active pressure applied in
this region. Generally, the pressure will become smaller at
the crown and increase laterally as the pressure arches over
the crown region locally. The gradual reduction in pressure at
21
3)
the crown during deformation of the lining will reduce the
final crown moment below that which would have occurred if
arching at the crown had not occurred or if the pressure at the
crown had been following.
Before the excavation 1S made, there 1S a vertical 1n situ
stress at the sides of the proposed opening that results 1n an
1n situ horizontal stress due to the Poisson effect equal to
v/(1-v) times the vertical stress if the medium 1S assumed to
be elastic or K times the vertical stress if the coefficient o
of earth pressure is known. This horizontal pressure makes the
all-around pressure on the lining more uniform and therefore
decreases the moments by decreasing the ovaling. This pressure
is in addition to the lateral pressure described 1n first
component listed.
4) The lining is generally less stiff overall than the medium it
replaced (especially if the deformation of the medium 1S
considered before ground-liner contact 1S made). Therefore,
load above the lining is arched around the sides increasing the
vertical stress in the spring1ine region and reducing the load
applied to the lining because the total overburden load is
resisted by the lining and by arching 1n the medium; the
division of load into the two parts depends on the relative
stiffness of the lining and medium. The change in vertical
stress 1n the spring1ine region in the medium due to arching
has an additional effect of increasing the horizontal pressure
on the lining due to the Poisson effect in this region.
Thus, this positive arching has two beneficial interaction
components-reduction of active vertical load on the lining and
increaseof horizontal passive pressure at the spring1ines.
1.3.2 Closed Form Solutions
Ranken, Ghaboussi and Hendron (1978) and Einstein et. al. (1980)
provide a good discussion of the history and applicability of the
22
closed form solutions for a lining embedded in a homogeneous medium.
These solutions are based on two-dimensional, plane strain, linear
elasticity assumptions in which the lining is assumed to be placed deep
and ~n contact with the ground (no gap). Early solutions by Burns and
Richard (1964), Dar and Bates (1974), and Hoeg (1968) were derived for
the overpressure loading described above, while solutions by Morgan
(1961), Muir Wood (1975), Curtis (1976), and Ranken, Ghaboussi and
Hendron (1978) were for the excavation loading. Solutions for both
loading conditions are available for the full slip and no slip
conditions at the lining-medium interface. The full slip solution will
give conservative (high) estimates of moment. The difference between
the slip and no slip conditions is not great for the overpressure and
excavation loading cases, but is large for the gravity loading and
loosening loading cases, particularly if vertical rather than radial
active loadings are assumed to act on the lining.
These closed form solutions contain all four of the interaction
components described above subject to the assumptions on which they are
based. They do not, however, allow for a gap to occur between the
lining and medium or for the lining to be placed at a distance behind
the excavation face. If the ground and lining remain linearly elastic,
the excavation loading is more reasonable than the overpressure loading
because the ground stress and deformation due to the in situ stress has
occurred when the lining is placed. In fact, more deformation must
occur when the excavation is made and before the lining and ground can
make contact so this case ~s an upper limit for the linear elastic
assumptions. However, if the medium deformations are large before the
lining and medium make contact and the shear strength of the medium is
exceeded (as in a soft plastic clay), time-dependent deformations may
continue for some time, relieving the stresses in the soil around the
opening and applying more pressure to the lining,(see Figure 1.2). In
this case, the excavation loading may no longer provide an upper limit
for thrust.
Design charts that provide dimensionless moment and thrust values
for various ratios of deformability of the medium to that of the lining
23
have been published for these solutions (Ranken, Ghaboussi and Hendron,
1978). Also, they provide a theoretically correct pa~r of
dimensionless parameters for comparing the properties of the medium to
that of the lining for the linear case. These parameters, called the
"compressibility" and "flexibility" (Peck, Hendron and Mohraz, 1972)
greatly facilitate the design process and provide insight into how
medium stiffness and lining properties affect the solution for moment
and thrust.
The solution for the excavation loading and full slip between the
medium and lining as given by Ranken, Ghaboussi and Hendron (1978) is
as follows with the symbols and sign convention defined in Figure 1.3:
v = - yHR {(l - K )(1 - 2J ) sin 2 e} 2 0 f
(1 + 2 v ) C L
m = 1 + (1 - 2v ) C f m
F + (1 - v ) J m
= 2F + (5 - 6 v ) f m
2 E
R (1 - v9,)
C =-..!!! (t)' [(1 2 v )] E9, + v )(1 m m
2 E (~)3
2(1 - v9, m
F -- [(1+v)] E9, t m
24
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In these formulas, the effective soil stresses should be used to
obtain the moment, thrust and shear, and then the thrust due to water
pressure added to that obtained from the formula for thrust. The water
pressure 1S omitted from the formulas because it does not result in
moment and shear.
1.3.3 Beam-Spring Model
The most important aspect of a realistic analysis is that it should
allow interaction between the medium at the sides of the opening and
the lining, because this stabilizing paSS1ve normal and tangential
pressure improves the strength and decreases the deformation and
cracking of the lining. If the lateral presence and its distribution
were known at the equilibrium position, it could be applied with the
loads, the lining analyzed, and a valid solution would be obtained.
However, the lateral pressure depends on the properties of the lining
(deformability) and the stiffness of the medium and 1S not known
beforehand in terms of its magnitude or distribution. In the past,
some design approaches have depended on a realistic estimate of this
lateral pressure, have assumed it to be uniformly distributed, and have
provided satisfactory designs. However, these methods do not provide
for different lateral pressure when the lining stiffness changes or
even when the medium stiffness changes in many cases.
An approach that would give a more realistic model of the
interaction when the load is applied directly to the lining would be to
replace the medium at the sides with radial and tangential springs that
have the same deformation characteristics as the medium and model the
lining with a series of linear beam elements. In this case, loads can
be applied in any location or shape and the medium properties can be
varied easily. The radial springs represent the radial passive
pressure on the lining and the tangential springs represent the shear
stress between the medium and lining. If the rock surface 1S smooth,
the tangential shear stress could be small but if the overbreak is
irregular as in drill and blast excavation, the shear stress will
depend on positive interlock and will be high. In the analysis, radial
26
( ,
springs that would be in tensions should not be allowed because it
would imply that the medium is pulling on the lining. In the ground,
the lining does not actually pull away from the mediQm, but there is a
relief of the passive radial pressure of the ground on the lining. The
tangential springs should remain active in the loaded region S1nce the
rock applying the loads would remain in contact with the lining and
apply tangential stress. Examples of this model are shown 1n Figures
3.14 and 3.44.
There are programs available that are designed for this type of
problem and automatically disconnect tension springs and therefore are
convenient to use. The model can also be constructed with any frame
analysis program using bar elements if spring elements are not
available; it will then be necessary to disconnect radial tension
elements through a trial and error process but the same solution will
be obtained once the radial tension elements are made inactive. In
addition, this model of behavior is suggested as a minimum analysis and
is not meant to exclude more complete models of the
those that represent the ground with continuum
interface between lining and medium with interface or
behavior
elements
joint
such as
and the
elements.
If these models are available, they can provide a more complete picture
of behavior. The load should be applied directly to the lining in
these models as suggested above, however, for the gravity or loosening
load.
Loads on the final lining are likely to require a long time to
reach their full value; this 1S particularly true if the load is
resisted fully by the initial support before the final lining is
installed. To account for this time effect on the deformability of the
concrete lining and the consequent effect on the interaction, it 1S
suggested that the modulus of the concrete used in the analysis to
obtain the beam element properties be reduced to one-half the value
g1ven by the ACI formula (ACI 318-77). Contractors often increase the
cement content of the specified m1X to improve pumpability and early
strength for form removal. In this case, the concrete compressive
strength and liner modulus will also increase. The actual 1n situ
27
liner modulus should be used if it can be determined, as it will reduce
the relative stiffness of the liner and medium and ~ncrease the
calculated moment. The contractor may also ~ncrease the lining
thickness by making the mined excavation larger than the specified
m~n~mum. This will decrease the flexibility ratio and increase the
lining moments. The effect of these variations on the design
parameters should be checked by the designer.
Radial springs used to represent the medium for a circular lining
may be given a stiffness based on the formula (Dixon, 1971),
K E be
m (1 + v ) r
m
where
K = radial spring stiffness representing the medium included by r
angle
E m
~n situ modulus of elasticity of the rock mass,
v ~n situ Poisson's ratio of the medium, m
b = length of lining under consideration ~n the longitudinal
direction,
e = angle subtended by tributary area of spring in rad.
The elastic modulus of the medium E for the rock mass must be reduced m
from the intact laboratory values of rock samples to account for the
effects of joints and shear zones. The amount of reduction depends on
the joint spacings, orientation, and fill material.
The following formula for modulus of subgrade reaction ~s suggested
for use to obtain the radial spring stiffness when arches are analyzed:
28
k
where
E m
2C o
k modulus of subgrade reaction,
C = arc length of the arch that 1S under compressive load from the o
footing to the point where separation of arch and medium
occurs (normally about one-third the diameter of the lining).
The resulting subgrade modulus must then be multiplied by the tributary
area for each spring. Therefore the spring stiffness would be
K r
k R b 8,
where
R = radius of the lining.
Tangential spring stiffness is more difficult to determine, but
studies show that they generally lie between 20 and 50 percent of the
stiffness of the radial springs depending on the surface between the
lining and medium. A value near 20 percent may be appropriate when
there 1S a smooth surface between the lining and medium while a value
near 50 percent may be more appropriate for a surface with irregular
overbreak. The effect of the ratio of tangential to radial spring
stiffness is discussed in Chapter 3 and some guidance to the most
likely value is given in Volume I of this report.
If the lining being considered is an arch with footings, then the
medium deformation under the footing should be represented by a spring
also. This spring should be based on the modulus of subgrade reaction
and can be calculated from the formula
K F
E m = Ie(C b) 1 1
= E b
m
2
29
where
KF stiffness of the spring that represents the footing
Cl
width of the footing
The number of beam elements needed to represent the lining depends
on the lining geometry and the desired spacing of springs (or elements)
to represent the medium. Springs should be placed at the joints at the
ends of each beam element and the springs should be close enough to
provide a smooth variation of tangential and radial passive resistance.
If the resisting pressure has a steep variation in some region, then
the beam elements should be made shorter to provide closer spacing of
the springs. In general, the angle subtended by beam elements in the
lining should be between approximately 7.5 and 15 degrees, with the
smaller value used when practical.
Section properties for the beam elements can normally be based on
the gross dimensions of the lining between node points. If the area of
present steel reinforcement is small, it can be ignored 1n calculating
the lining stiffness; if the area is large or if a composite section is
used, the transformed section should be considered in calculating the
section area and moment of inertia. The section dimensions should
include all the concrete that can reasonably be expected to be present
in the final structure. In some cases where cracking may be extensive
in a small region of the lining, it would be reasonable to use a
cracked or partically cracked moment of inertia for the elements 1n
this region.
When the beam-spring model is used, additional variables can be
considered that are not included in the closed-form solutions, such as
a nonuniform distribution of applied load, variable lining stiffness
around its circumference, joints in the lining, or variation of ground
properties in a layered media. It can be used for loosening loads in
rock or for gravity loading in soils. However, the linear analysis of
this type will always provide conservative designs because the strength
30
'0 ,.j~
" ."'
will be larger than that computed. In the next section, the difference
between the linear and nonlinear analysis 1S discussed for the
beam-spring model, and quantitative comparisons are made in Chapter 3.
Load 1S applied directly to the lining in the beam-spring model,
causing it to oval or deform outward at the springlines and causing a
resisting pressure from the medium that is proportional to the lining
deformation; thus the first component of interaction described in
Section 1.3.1, due to overall deformation, 1S satisfactorily modeled.
Since the load is applied directly to the lining, the second component
(change in shape of the active pressure due to local arching) 1S not
taken into account; however, if the shape of the applied load can be
estimated, it can be given the proper shape when it 1S applied.
Neither of the interaction components three or four are automatically
considered, but they can be included approximately if they are
considered to have a significant effect; this would be done by applying
horizontal pressure to the lining just as the vertical active pressure
1S applied to represent the medium pressure due to the Poisson effect
and by modifying the vertical load to account for overall arching
around the lining.
1. 3.4 Nonlinear Analyses
When the loads on a lining become large enough to cause the
concrete to crack and at higher loads cause the concrete compressive
stress to become nonlinear or the reinforcement to yield, the lining
becomes less stiff at these locations and the moment is reduced or does
not continue to increase. The overall stiffness of the lining relative
to the medium is reduced and the lining deflects at a greater rate than
before these events occurred. The resulting additional deformation
causes an increase in the passive pressure of the medium at the sides
and causes the pressure around the lining to become more uniform;
consequently, more of the additional load is resisted by a pure thrust
mode in the lining.
31
When cracks form and there 1S limited tension at the section
because the reinforcement yields or there is no tension in unreinforced
linings, the resistance to thrust is similar to that 1n an unbolted,
segmented lining or brick arch, except that the lining is likely to be
thinner than these structures that are designed as pure compression
arches and proportioned so that the thrust remains within the center
portion of the section. Cracking implies that no stress occurs over
part of the section and so the area in compression 1S reduced. As
deformation of the lining increases the unstressed area increases and
the area in compression decreases. During this process, the resisting
moment does not increase very much with increasing load, but the thrust
may increase significantly. A load is finally reached in which the
increased thrust and decreasing compression area result in compression
failure of the concrete; this failure is closely related to the lining
deformation because it is related to the deformation of the failure
section and thus the area of concrete 1n compression; the lining
deformation is closely related to the stiffness of the medium around
the lining.
The load at which failure finally occurs is larger than that which
would be predicted by a linear analysis and the difference depends
largely on the relative stiffness of the medium and lining, which can
be indicated approximately by the flexibility ratio. The difference is
larger at small values of flexibility ratio and decreases as the
flexibility ratio 1ncreases. For large values of flexibility ratio
that result for linings 1n rock, the linear analysis is probably
adequate for design, but for linings in softer materials it may be more
economical to take advantage of the nonlinear increment of strength.
There are several general purpose programs available that will
perform nonlinear analyses of structural frames and can be used for the
lining analysis. A program developed for this project is based on
using a series of straight beam elements for the lining that use the
nonlinear concrete and reinforcement stress-strain curves to describe
the beam behavior and include the effect of geometric nonlinearity by
continually updating the coordinates of the lining. This program
32 .'
allows the concrete stress-strain curve to have a descending branch
after the peak strain, which is necessary to predict the peak lining
load. Most other nonlinear programs do not allow the descending branch
in describing the material properties.
This same lining representation 1S used with radial and tangential
springs to represent the medium (the beam-spring model) or it is used
with continuum elements to represent the medium (the beam-continuum
model). The springs in the former model and the continuum elements in
the latter remained linear. However, an interface or joint element can
be used between the lining and medium in the beam-continuum model that
has limited shear properties described by cohesion and angle of
internal friction parameters. If the beams are given linear properties
in this model and the interface elements are given full slip or no slip
properties, it provides essentially the same results as the closed form
solutions when the same loading is applied (Section 3.3 of Volume I).
There are small differences resulting from the discretization of the
lining and the medium that become smaller as the beam and continuum
elements are reduced in size.
The reason for using the beam-continuum model of this type is that
it offers versatility in working some types of problems that cannot be
handled by other solution methods. The beam-continuum approach allows
consideration of depth of cover, noncircular lining shapes, partial
slip as well as full and no slip between the lining medium, a
nonuniform or stratified medium, nonlinear behavior of the lining and
nonlinear interface stresses between the medium and lining. Also, the
properties of the lining can vary around its circumference to account
for cracking, joints, or a variation in design which the closed form
solution cannot handle. It also has all the advantages of the
beam-spring model but in addition, the excavation and overpressure
types of loading that result from the combined deformation of the
lining and medium can be handled. Therefore, there are certain
worthwhile
desirable if
includes all
advantages to a beam-continuum solution that may be
the design problem warrants their use. This model
four components of interaction described above. It will
33
also provide the additional strength of the lining that results from
nonlinear lining behavior and the consequent redistribution of moment.
It will not allow a void to occur between the lining and medium before
contact between the two occurs, nor will it allow the medium to have
nonlinear behavior
incorporated.
unless nonlinear
34
continuum elements are
:.'
CHAPTER 2
EVALUATION OF EXISTING DESIGN PRACTICE
2.1 PURPOSE AND METHODOLOGY
Sixteen design firms that are engaged 1n tunnel design of various
types were interviewed to survey the existing procedures used for
design of underground concrete structures. This survey served the
purpose of: (1) providing insight into the existing design methods,
(2) showing where uncertainty exists 1n the design procedures and
therefore where research would be most fruitful, and (3) indicating
how new developments can be formulated to fit into the existing design
framework.
Initially, the literature was surveyed in an effort to determine
the procedures used by designers, but it became clear that most of the
reported information concerns the analysis techniques used and many of
the assumptions and simplifications and their implications on the
design are not discussed. Consequently, it was decided to conduct a
survey by visiting var10US design firms around the United States in the
fall of 1980. The information described in this report is derived from
discussions with the tunnel designers. Since the aim of this research
effort is to propose design guidelines for transportation tunnels, most
of the designers interviewed were engaged in this type of work but a
few designers of water conveyance, hydroelectric, and sewer tunnels
were also consulted to broaden the scope of the study and to show the
influence of the function of the tunnel on the design procedures.
Furthermore, because of the intimate relationship between the design
and construction processes, the contractor's point of view was taken
into account
organizations
tunnels.
by
that
visiting representatives
are currently engaged 1n
of
the
two contracting
construction of
There is a great diversity of design procedures used, even more
than expected at the outset. It may be said that no two firms use the
35
~.'
same procedures for design, though there may be some similarities ~n
part, such as determination of the loads or the type of analysis used.
For this reason, a summary is rather difficult. Also, there is always
some confusion during the discussion concerning what the firm actually
does and what the representative feels they should do if they had the
time or money or were working under ideal conditions. There ~s
sometimes a lack of agreement among individuals in the same firm. In
some cases, the views of the representative have been changed by the
last job, and the procedure used on the next job would be different.
An effort was made, however, to determine an overall design philosophy
used by the firm.
The firms and individuals visited were selected because of their
involvement with recent tunnels and are listed alphabetically in Table
2.1. It was not possible to contact all tunnel designers in the United
States, but it ~s believed that those interviewed constitute a
representative group. An effort has been made to report as accurately
as possible the design approaches that were discussed in each case
strictly to compare design methods and not to criticize or judge any
procedure.
A brief summary of the findings from the interviews is presented ~n
the remainder of this Chapter, and more detailed descriptions may be
found in Appendix A of Volume I. For conven~ence ~n the discussion,
the design approaches are divided into loading, analysis, and criteria
for strength and serviceability. Within each of these categories, the
discussion is further divided into running tunnels in rock with a
diameter of 30 ft (9 m) or less, large openings in rock with spans of
40 to 60 ft (12 to 18 m) and running tunnels in soft ground. The main
reason for the distinction between opening sizes in rock is the effect
of opening size, method of excavation, and type of initial support on
the loading of final linings.
36
Firms
1. Bechtel (San Francisco, CA)
2. DeLeuw Cather (Washington, DC)
Bechtel (Bethesda, MD)
3. Denver Board of Water Commissioners (Denver, CO)
4. D.M.J.M./K.E. (Baltimore, MD)
5. Goldberg-Zoino Associates of NY (Buffalo, NY)
Hatch Associates (Buffalo, NY)
6. H & A of New York (Rochester, NY)
7. Harza Engineering (Chicago, 11)
8. Jacobs Associates (San Francisco, CA)
9. Jenny Engineering (South Orange, NJ) (Milwaukee, WI)
TABLE 2.1 LIST OF PARTICIPANTS
Designers Intervie,ved
Walter Ferris Art Arnold Chris Gardner
Dah Fwu Fine Kuldip Singh
Carl Bock
John Parsons Verne Hornback Jim Batt W. Colwell, Jr. (with DMJM PhillipsReister-Haley, Inc.)
David Hammond Drupad Desai
Richard Flanagan
Wesley Terry
Gary Brierley
Jan Veltrop Arvids Zagars William Bristow William Shieh Jerry Hahn Ed Cikanek
James Wilton
Prakash Donde Lloyd Monroe
37
Recent Tunnels Discussed
Sultan Tunnel, Washington State (Designer). Also acting as construction manager in various tunnel projects.
Washington Metro (General Engineering Consultant)
Washington Metro (Construction Consultant)
Foothill Tunnel, Roberts Tunnel, Arrow Tunnel, Colorado
Baltimore Region Rapid Transit System
Buffalo Light Rail Rapid Transit. (Geotechnical Consultant) Sewer Tunnel in Rochester
Buffalo Light Rail Rapid Transit. (Principal Engineering Consul tan t , Rock Tunnel Section)
Sewer Tunnels in Rochester, NY Portions of Boston Subway
Chicago's TARP Project Zoo Park, Cleveland Park and Van Ness Stations in Washington Metro and tunnels and shafts for various hydroelectric projects throughout the world.
Primary support for San Francisco sewer tunnel, Washington Metro (section D-9)
Milwaukee sewer tunnel (Soft Ground) Sewer in Rochester (Rock) Sewer in Bankong (Concrete Segments in .Soft Ground) Redesign sewer tunnel in Detroit
TABLE 2.1 LIST OF PARTICIPANTS (Continued)
Firms
10. Leeds, Hill and Jewett, Inc. (San Francisco, CA)
11. New York City Transit Authority (New York City, NY)
12. Parsons-Brinckerhoff (New York City, NY)
13. Parsons-Brinckerhoff/ Tudor (Atlanta, GA)
14. Tudor Engineering (San Francisco, CA)
15. U.S. Army Corps of Engineers; Missouri River Division (Omaha, NE)
16. Bureau of Reclamation (Denver, CO)
Designers Interviewed
John Bischoff
Mel Oberter Abe Blumberg Richard Mitchell
Tom Kuesel Dan Wallace Bill Daly Bill Hansmire
Douglas J. Mansfield Zdenek Zachar Perry M. Lin
Don Rose Heinz Mueller Richard Mayes
Dan Hokens
Paul Tilp Timothy Smirnoff Ken Schoeman
38
Recent Tunnels Discussed
Eisenhower Tunnels, Colorado
New York City Subways
BART System, Lexington Mkt. section in Baltimore WMATA (F2A,C4), Atlanta Subway (N120)
Atlanta subways
San Francisco cross town tunnel
Primarily water conveyance tunnels
Hades and Rhodes tunnels, Utah Stillwater tunnel, Utah
2.2 LOADING
2.2.1 Tunnels ~n Rock
Most running tunnels in rock are supported temporarily until the
excavation ~s completed and then the final lining is installed in the
stable opening. Ten of the firms visited select the final lining on
the basis of minimum thickness for convenient construction and do not
make calculations based on ground loads. Some designers reason that
the temporary support has stabilized the opening and the final concrete
lining is only needed to maintain that stability, to provide leakage
control or to fulfill other owner requirements. Others reason that the
capacity of a lining with nominal thickness is more than adequate for
most cases, even if loads were applied. It should be pointed out that
some of the firms who make this assumption design sewer or water
conveyance tunnels ~n which case the s~ze may be smaller than
transportation tunnels and the owner requirements less strict.
Two firms who primarily design transportation tunnels select
typical rock wedges based on the site geology for loading
determination, while three others use an index of rock quality such as
the RQD (Rock Quality Designation), RSR (Rock Structure Rating) or
Terzaghi's loads or some combination of these for the design. One uses
the effective overburden and designs the lining to resist this load in
thrust. In these cases the loads are used to design the final lining,
and the initial supports are to a large extent left to the discretion
of the contractor.
All six of the design firms who select rock loads also consider
full or partial water pressure acting on the lining in conjunction with
the rock loads. In addition, six of the firms who assume that a
minimum lining ~s adequate, do check for water pressure loading while
the rest assume that it ~s adequate for external water pressure as well
as rock loads. One firm drains the tunnel whenever possible and does
not consider water pressure at all in those cases. A separate analysis
for internal pressure is always performed if it occurs. It is not
39
clear to what extent the other firms consider drainage, but those who
mentioned partial water pressure are probably considering that some
drainage occurs. Noncircular tunnels with long straight walls are more
likely to be drained to avoid stress concentration at the corners. In
view of the recent problems with clogging of some of the drains 1n
Washington, D.C. and Baltimore subways,
loading philosophy is uncertain at this time.
.are rarely (if ever) considered.
the whole water pressure
Dynamic and live loads
Ten of the firms had designed large openings 1n rock and obviously
their philosophy varied with the type of rock and function of the
opening. Two of the firms who have designed power plant chambers could
select the site and orientation of the opening to be most favorable and
so they felt that the need for a final cast-in-place concrete lining is
questionable with proper initial support. This could also be argued
for transportation tunnels in competent rock (i.e., Atlanta subway),
though the function may introduce other considerations such as
aesthetics, water seepage, ownership and legal requirements. Five of
the firms who have recently designed subway stations in less competent
rock (i.e., Washington, D.C. subway) based loads on local geology and
the selection of rock wedges or full overburden. Two firms have based
their design loads on the RQD concept or some modification of it, while
two others have used a minimum loading on the final concrete lining
that they base on the excavation and initial support sequence. They
advocate controlling these activities carefully so that the initial
support actually performs the primary support function.
Of the S1X designers who select rock loads based on rock wedges or
RQD, four of them consider full or partial water pressure with the rock
loads. Four designers prefer to drain the external water pressure,
while the remaining two would either design for water pressure or drain
the water depending on the site conditions and function of the opening.
40
2.2.2 Tunnels ~n Soft Ground
Six design firms provided information on loadings for tunnels ~n
soft ground of which three designed transportation tunnels and three
sewer tunnels primarily. Of the transportation tunnel group who
designed cast-in-place final concrete linings, one used a
overburden vertical load accompanied by a horizontal pressure of
full
0.875
times the vertical pressure for a working stress design and then the
lining ultimate strength was checked with a reduced lateral pressure.
Another used 1.5 to 2.0 diameters of soil weight above the crown for
the initial support and 60 percent of this value for the final lining
and used an analysis that represents the soil with spr~ngs, so the
lateral pressure depends on the soil stiffness. The third in this
group prefers to carefully control the excavation and initial supports
and then consider the m1n~mum constructable final lining to be
adequate.
Of the group who designs cast-in-place sewer tunnel linings, the
first uses full overburden reduced by soil shear resistance if the
final lining ~s installed within a few days of the initial support, and
full overburden if it is installed later. Lateral pressure near the at
rest earth pressure is applied. The next designer uses full overburden
up to 80 ft <24.4 m) depth and some portion of full overburden above
this value with a lateral pressure between the passive and at rest
earth pressure values. Alternatively, he may determine moments by
imposing a diameter change on the lining that depends on the tunnel
size. The last designer ~n this group considers the minimum
constructable lining adequate if care is exercised in providing
flexible initial support.
Five design firms provided information on segmented precast lining
design and all advocated first proportioning the segments for handling
and jacking loads and then checking for ground loads not greatly
different from those used for cast-in-place linings or checking for
thrust due to full overburden and moments due to a specified diameter
change. They emphasized, however, that care must be exercised in the
41
installation to obtain good contact with the ground through competent
backpacking and grouting without appreciable ovaling.
Most of the designers include the effects of external water
pressure and internal water pressure as well, when present.
2.3 ANALYSIS
2.3.1 Tunnels in Rock
Nine firms consider the minimum constructable cast-in-place lining
in running tunnels adequate and do not perform an analysis, and of the
seven remaining, two of them use a frame analysis in which they
represent the lining by a series of beam elements and the medium by
springs, and one uses a similar frame analysis for the lining and
finite continuum elements for the medium (these are larger than normal
running tunnels, however). The remaining designers use either closed
form solutions for a cylinder embedded in an elastic medium, design
charts and experience or a design for thrust based on full overburden
and moment based on an eccentricity that is some portion of the radius
of the tunnel.
Eight of the firms described their analysis for large open1ngs 1n
rock and emphasized that the geology at the site and function of the
opening influenced the extent of analysis required and even whether an
analysis is necessary. Five of the firms who designed large openings
used linear beam elements for the lining and four of them used radial
and tangential springs for the medium, while the other used plane
strain two dimensional continuum elements 1n modeling the rock.
Another design group uses the linear beam element for the lining but
does not allow the rock to deform at the sides of the opening. When
springs were used for the medium, the value of tangential spring
stiffness varied from zero to 50 percent of the radial stiffness. The
remaining firms perform simple analyses to check the lining strength
and depend primarily on experience and control of the excavation and
initial support sequence to assure stability of the opening with
42
. .
minimum final support; in some cases the ground and excavation and
support sequence were modeled with a finite element program to assure
stability during excavation and help determine amount of initial
support needed.
2.3.2 Tunnels in Soft Ground
Five firms provided information on their design approach to linings
in soft ground. Three firms use linear beam elements to represent the
lining, radial springs for the medium that are detached if they are 1n
tension, and do not include tangential springs. Two of this group also
include a lateral pressure to represent soil stresses acting in
addition to that from deformation of the lining and when this is done,
full overburden or nearly full overburden is used as the vertical load.
Another design group does not include the additional lateral pressure
but the vertical load is limited to 1.5 to 2.0 diameters of soil, the
load depending on the type of soil present. Full or partial overburden
soil pressure applied uniformly around the lining to obtain the thrust
and a predetermined diameter change to calculate the corresponding
moment 1n the lining is used by another design organization. A
variation of this method is also used in which the thrust is obtained
using the same approach but the moment is computed by applying this
thrust with a predetermined eccentricity that is some portion of the
lining radius. Another firm uses the closed form solution without
shear stress between the medium and lining presented by Peck, Hendron
and Mohraz (1972).
Those designers who use the beam-spring model for cast-in-p1ace
linings use the same model for segmented linings and reduce the
stiffness of the joints to zero or a low value. Both firms who design
for a uniform soil pressure and a moment from a fixed diameter change
or eccentricity also use the same general approach for segmented
linings. One firm uses an equivalent cracked section stiffness (EI) to
represent the joints in the lining. In all cases, the segments are
first designed for jacking forces, handling stresses and perhaps
43
grouting pressure, and these considerations usually govern the design.
One of the firms also checks stresses in the skin of the lini~g between
ribs as an edge supported plate subjected to uniform pressure; they
find that this check may frequently produce the largest stresses.
2.4 DESIGN CRITERIA
The strength criteria are not independent of the analysis used or
loads selected, but for purposes of comparison they will be divided
into the working stress method, in which actual expected loads are
selected and stresses in the lining are limited to some portion of the
material strength, and the ultimate strength method in which the
working loads are multiplied by load factors and the ultimate strength
of the lining is considered.
Thirteen firms described their strength criteria and of these seven
used working stress methods and four used ultimate strength methods
while two used both. One of these latter two firms uses either
approach depending on the owner's requirements; the other proportions
the lining using working stress methods and then checks it for ultimate
strength with reduced lateral pressure for running tunnels and with
increased water pressure for stations. Working stress values vary from
(0.25) f to (0.45) f for the concrete and (0.5) f to (0.66) f for . c c f d . l' Y h dY . the re1nforcement. Load actors use 1n the u t1mate strengt eS1gn
method vary from 1.2 to 3.0 depending on the load type.
Serviceability criteria in terms of leakage and crack control were
emphasized by the designers involved in transportation and sewage
tunnels, while it 1S of little concern for designers of water
conveyance tunnels. Most designers agree that for mass transit
projects, leakage specified by an allowable leakage criterion can be
tolerated in running tunnels, but it should be near zero for stations
without an interior lining. Cracking in linings is acceptable as long
as the leakage criteria are not exceeded. Nearly all designers specify
the use of a grouting program to seal cracks that do occur but other
measures to control cracking and leakage vary. Some advocate use of
44
drains, whereas others use reinforcement to limit cracking and the
resulting leakage.
There is wide disagreement among designers when it comes to the use
of reinforcement in concrete tunnel linings. All the designers of
transportation tunnels except two advocate the use of reinforcement in
the circumferential and longitudinal directions for both soil and rock
tunnels, but some use two layers and others only one. One designer
typically uses one layer on the tension face for linings in rock and
two layers for linings in soil. Most of the sewer tunnel designers use
one layer of reinforcement. The water conveyance tunnel designers use
no reinforcement at all, except in unusual ground conditions or when
the tunnel is pressurized. Some designers feel that some reinforcement
is necessary to prevent chunks of deteriorated concrete from falling
out, especially for larger diameter tunnels, and to assure the
long-term integrity of the lining.
Construction related criteria vary, depending on site conditions,
construction practices in the area, owner requirements and so on. Four
of the transportation tunnel designers specified lengths of casts that
varied from 20 to SO ft (6 to 15 m) with vertical construction joints
to control shrinkage cracking. Several indicated that they thought a
sloping joint at the angle of repose of the concrete would be more
economical and would not cause any more leakage or other problems. The
contractors were convinced that sloping joints, if properly made would
cause less leakage problems. Some designers specify an interval, in
terms of time or distance, between excavation and placement of a
cast-in-place lining to assure that the opening has stabilized with the
initial supports to avoid cracking of the final lining.
2.5 AREAS OF UNCERTAINTY IN DESIGN
Reinforcement: The greatest uncertainty among designers appears to
concern the use of reinforcement in the final concrete lining. Many
designers advocate the inclusion of reinforcement 1n the final lining
without a valid justification. The argument that it contributes to the
45
long-term integrity of the final lining used by some designers 1S
challenged by others saying that reinforcing bars corrode with time,
increase in volume and thus cause spa11ing in the concrete, 1n essence
causing deterioration rather than preventing it. In many cases it is
not clear whether reinforcement is needed to resist ground loads, to
control shrinkage cracking or both. Even in the case when it is
decided that reinforcement is needed, there is uncertainty whether to
use a double or a single layer, closer to the inside face or in the
center of the section. There is general agreement, however, that costs
could be reduced significantly if reinforcement could be left out,
except 1n those cases where it is definitely needed.
External Water Pressures: Handling of external water pressure on
linings is also uncertain in terms of when to design for it and when to
drain the area around the lining to reduce the pressure. There is a
tendency to try to drain large openings in rock and tunnels with sharp
angles or long straight walls. In these cases, there 1S the
possibility of high external hydrostatic pressures creating stress
concentrations or high tensile stresses.
However the decision to drain does not necessarily resolve the
1ssue. Recent problems with clogged drains behind some tunnel linings
in Washington, D.C. and Baltimore subways indicate that the question is
not whether the tunnel ought to be drained but whether it can be
drained over the entire life of the tunnel. Therefore, careful
analysis of the chemistry of the ground water and provisions for
maintaining the drain openings were mentioned as important design
considerations.
Loading for Linings in Soft, Ground: The diversity of approaches
used 1n the design of the final concrete lining for tunnels in soft
ground indicate a lack of agreement among engineers. Most designers
agree on the concept of placing the initial flexible support and
allowing the opening to deform until it is stable before the final
cast-in-p1ace concrete lining is placed. Opinions vary on the loading
that then occurs on the final lining. The magnitude of the ground
46
loads applied over the crown are usually related to the nature and
magnitude of the assumed side reactions. When full overburden ~s
selected as the vertical load, a fairly large portion of the vertical
pressure is applied horizontally; when the vertical load is limited to
1.5 to 2 diameters of soil weight, lateral pressure is assumed to occur
only due to deformation of the lining.
Analysis of Precast Segmented Concrete Linings: Because of the
lack of experience with precast segmented concrete linings in this
country, there is uncertainty concerning the approach that should be
used for their design. It is generally believed that designs based on
handling, erection and jacking forces will also be adequate for ground
loads. It was pointed out, however, that the joints are sometimes
significantly weaker than the cross-section of the segment, depending
on the joint details, and in some cases may become a critical section
under ground loads.
Cracking and Leakage: Cracking of the final concrete lining, per
se, is not considered to be a problem unless it is related to leakage.
Some designers consider the tunnel lining satisfactory even if cracked,
as long as it does not leak. In sections of transportation tunnels in
particular, where people are present, leakage control is considered
essential. Unfortunately, the designer is not in a position to
guarantee a dry tunnel during the design stage, unless a well proven
drainage system, or a shield interior lining is used. Certain
designers attempt to specify "low" stresses in the longitudinal
reinforcement to control load-related cracking and minimum amounts of
longitudinal reinforcement to control shrinkage-related cracking. A
lot of designers feel, however, that instituting a post-construction
crack treatment program is the most effective way to control leakage.
Degree of Conservatism ...!!!. the Design 2! Larger Openings In Rock: As
the size of a tunnel opening in rock increases, so does the degree of
uncertainty with regard to the loads that will reach the final lining
and the degree of conservatism in the overall design. The possibility
of large rock blocks sliding along discontinuities in the rock mass
47
(joints, shear zones, etc.) and applying relatively concentrated or
eccentric loads does exist. Furthermore, the excavation and initial
support sequence become extremely important in determining the
magnitude of rock loads that are likely to reach the final lining.
Also the consequences of problems during construction and after the
facility is in use have far reaching financial implications. For these
reasons, the designer generally is very conservative in his approach.
Effect ~ the Initial Support System ~ the Final Concrete Lining:
Different philosophies exist as to whether to account for the initial
support system or not and if it is considered, how much strength it
provides to the lining system or how much it reduces the load on the
final concrete lining. If the initial support system is considered,
then the loads on the final concrete lining are usually reduced by an
amount which depends on the type of the initial support used.
The research effort was directed toward providing answers to these
areas of uncertainty in the design of the final concrete linings. Not
all questions have been answered, however, since some of the existing
uncertainties will be resolved only through long-term field
measurements, and their correlation with analytical and experimental
results.
48
CHAPTER 3
SUMMARY OF RESEARCH AND DESIGN IMPLICATIONS
The research consisted of a series of model tests on arch and
circular linings and development of a finite element analysis that was
used to study effects of parameter variation on arch linings in rock
and circular linings 1n rock and in soft ground. Volume I contains
detailed descriptions of these studies as well as background
information and implications on design. A summary of the results and
their design implications is presented in this chapter to provide the
reasoning behind many of the design recommendations made in Chapter 4.
3.1 MODEL TESTS
3.1.1 Arch Linings In Hard Medium
Test Description: Two types of models were tested to evaluate
certain behavioral characteristics of arched linings in rock. In
particular, the effects of loading shape, tangential shear between
lining and medium, flexibility ratio ahd lining reinforcement on the
overall behavior, and failure mechanisms of the lining were evaluated.
It 1S reasonable to compare the effects of each variable among these
tests when all other variables were the same; though the scaling of
information to the full scale structure in the ground, discussed in
Section 2.3 of Volume I, may be questioned, scaling can still give some
idea of the effects of the same variable in a full scale system. The
results and conclusions from these tests will be summarized in this
section.
Ten arches, 6 ft (1.8 m) in diameter and varying thickness,
surrounded by a concrete medium to represent rock, were tested. They
were loaded by applying forces with hydraulic rams directly to the
upper 60 deg segment centered on the crown to simulate loosening loads.
The test arrangement and loading system are shown in Figure 3.1a. Two
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medium stiffnesses were used, first by using only the concrete and
second by replacing some of the concrete by neoprene pads; the actual
in-place effective medium stiffness was determined with plate load
tests. Shear deformations between medium and lining were prevented 1n
some tests by casting serrations in the medium as shown in Figure 3.lb
and in one other shear stress was reduced by providing a smooth
lubricated surface. The hydraulic ram forces could be varied to
maintain the load shapes shown in Figure 3.2.
Table 3.1 summarizes the variables and Table 3.2 provides a summary
of the most significant test results. In these comparisons strength 1S
normalized as the ultimate thrust in the lining at failure T divided by u
the ultimate axial thrust or that which would occur if there was no
moment, T The reason for this normalization 1S that the strength o
ratios can be compared among tests when the concrete strength is
different or the reinforcement is present or not, in order to determine
the effects of other variables. Reduction of this ratio from the value
of one is due to the presence of moment at the failure section, and can
be viewed as a relative strength without regard to the actual load and
how it occurred. The strength ratio 1S shown on Figure 3.3 as a
function of the flexibility ratio F for all the tests. The flexibility
ratio has been derived as a measure of equivalent stiffness of the
medium to that of the lining in flexure for a circular lining (Peck,
Hendron and Mohraz, 1972) and the formula is given in Section 4.3.4.
This derivation 1S not applicable to arches directly, but it still
provides a convenient measure of this relative stiffness and contains
the appropriate variables; therefore, it is used here as the variable
describing the relative stiffness of the lining-medium system.
Effect of Load Shapes: The load shape that gave the lowest
T IT ratio at F = 1200 and interlocking between the lining and medium u 0
was the symmetrical triangle with load concentrated at the center. The
largest T IT resulted from the uniform load. The range was from 0.60 u 0
to 0.69. However, arching in the field is likely to limit the
magnitude of nonuniform loadings more than those that are uniform,
making the uniform load the one that gives the lowest safety factor
51
Idealized Geologic Condition
Rigid Rock Block
Rock Joint
Corresponding Loading Pattern
1111111
FIGURE 3.2 LOAD SHAPES USED IN ARCH TESTS
52
Vi
W
i I
Des
ign
atio
n
ARC
H-1
ARC
H-2
AR
CH
-3
AR
CH
-4
AR
CH
-5
AR
CH
-6
AR
CH
-7
AR
CH
-8
AR
CH
-9
AR
CH
-10
TABL
E 3.
1 SU
MMAR
Y OF
ARC
H TE
STS
IN
HARD
MED
IUM
-T
hic
kn
ess
Roc
k r·l
odul
us
Tan
g.
Sh
ear
Rei
nfo
rced
L
oadi
ng
Sha
pe
psi
(l
ira)
[~ll~l!
3(7
6)
1.68
x10
6 (1
1575
) YE
S NO
~
3(76
) l.
68>
: 10
6 (1
1575
)
YES
NO
rr1Yr
ll 3
(76
) l.
68;.
:10
6 (1
1575
) YE
S rio
~
3(76
) 1.
68x1
06
(115
75)
YES
NO
II1III
l 1
.68
xl0
6 I
3(7
6)
(115
75)
NO
NO
Ij t
f ttl
R
IGID
3
(76
) l.
68>
: 106
(1
1575
) YE
S rio
It .... "4
+ I
RIG
ID
3(7
6)
0.1
7x
l06
( 117
0)
YES
rIO
11111I
1 3(
76)
0.
17x1
06
( 117
0)
YES
NO
1111II1
1
(25)
0.
17x1
06
( 117
0)
YES
NO
111111
1 1
(25)
O
.llx
lO6
( 117
0)
YES
YES -
-_
.-
VI ~
..
Tes
t :
Des
igna
tion
I
ARCH
-1
ARC
H-2
ARCH
-3
ARCH
-4
ARC
H-5
ARC
H-6
ARCH
-7
ARC
H-8
ARC
H-9
ARC
H-1
0
TABL
E 3.
2 SU
MMAR
Y OF
ARC
H TE
ST
RESU
LTS
F1ex
ib
Tang
T
hrus
t L
oadi
ng
Shap
e R
'ltio
S
hear
Re
in f.
nH
~
IIIIII
I 0.
69
~
0.60
YE
S
lfNrll
12
00
0.67
~
NO
0.64
IIllrr
l NO
0.
52
It t ,.,
. 1\
RIG
ID
0.64
Ii; * i
t I
RIG
ID
0.37
12
0
IIIIII
I YE
S 0.
42
II r r 1
I I
0.62
3650
II I r 1
r I YE
S 0.
65
Cra
ckin
g Lo
ad
1'1 Rj
Ws
:; of
Ult
imat
e
0.53
NO
CR
ACKS
0.61
50
0.41
NO
CR
ACKS
0.52
NO
CR
ACKS
1. 0
9 80
0.30
~j
Q CR
ACKS
1. 5
0 40
~
1. 0
3 60
1. 0
3 85
1. 6
0 80
0.9
0.8
0.7
0.6
~
........ ~
0.5
0 \.J
l +
-\.J
l 0
0.4
a::
+-
IJ)
::l
0.3
... .c
I-
0.2
0.1 00
x
Arc
h-I
A
rch
-3
Arc
h-I
O
(Ful
l In
terl
ock
) [}
d]
[[I]
J (R
ein
forc
ed
)
ITilJ
" I
4 •
Arc
h-9
--
--~;
::;:
:;-. /~
---
'-f1
.. A
rch
-_
__
__
__
_ •
[[I]
J (
Un
rein
forc
ed
) Ar
ch-6
~...
...
Arc
h-2
_
-----
.~~------
----
--)x
t ~
Arc
h-5
(0
.33
---
·'CII
IJ (F
ull
Slip
) ,,
,..
."
...... [
]]]]
Arc
h -8
• ffi
tE A
rch
-7
20
00
3
00
0
12
0
36
50
Fle
xib
ilit
y R
atio
, F
FIGU
RE 3
.3
EFFE
CTS
OF L
OADI
NG S
HAPE
AND
FLE
XIB
ILIT
Y
RATI
O ON
THE
THR
UST
RATI
O
"
against failure. At F = 120 loads were applied uniformly and from a
rigid block, and the T /T values were 0.42 and 0.37. Failure of the u 0
specimen with triangular loading that peaked at the crown was ductile
and resulted from flexure at the crown. Comparable tests with other
shapes of loading failed more suddenly and at or near the edges of the
loaded area. Load shape also had an effect on the deformability of the
lining with the rigid block loading (Arch-6) exhibiting the smallest
ductility as shown in Table 3.2.
Effect of Tangential Shear Between Lining and Medium: The effect
of tangential shear is shown by comparing Arch-l with full interlock
and Arch-5 with no interlock, and both with the same F of 1200.
Removal of interlock reduced T /T from 0.69 to 0.52. Removal of u 0
tangential shear leads to larger deflections, which allows the crown
region to flatten more and creates a larger moment; it also increases
greatly the force reaching the base of the arch. With interlocking the
base force was from 5 to 30 percent of the total load, but without
interlock this base force was slightly over 100 percent, a sizable
increase as far as the design of the base footings is concerned.
Effect ~ Flexibility Ratio: The rock modulus and lining stiffness
are incorporated in the flexibility ratio which has a marked effect on
T /T as shown in Figure 3.3. There is a problem with the comparison, u 0
because at F = 1200 the full interlock and no interlock cases were
tested, while at F = 120 and 3650 neoprene was used between the lining
and medium which would be equivalent to a partial interlock between
these extremes. By trial and error, in trying to match the analysis
with experimental results, it was determined that these latter cases
with the neoprene corresponded to a tangential shear stiffness that was
about one-third of the radial stiffness, so the curve shown in Figure
3.3 is drawn 1/3 of the way between Arch-5 (no interlock) and Arch-l
(full interlock) at F = 1200. The resulting curve is then for the same
tangential shear condition and uniform loading. Though exceptions can
be taken with the value used, the curve appears reasonable, and for
this loading case T /T varied from 0.42 at F = 120 to 0.62 at F = u 0
3650. Also, the curve is rather flat beyond F = 2000, indicating that
56
_ ..
there is an upper limit for T IT , which is about 0.62 for the uniform u 0
load case 1n good quality rock. This conclusion is based on the
assumption made at F = 1200 to draw the curve, but any assumption made
will result in a fairly flat curve. It may reasonably be expected that
there is a similar limit for other load shapes or other tangential
shear conditions. A minimum amount of moment should be expected in the
lining no matter how high the value of rock modulus. This moment
causes the thrust ratio to drop from 1.0 (no moment) to about 0.62 for
these test conditions.
Cracking: Cracking characteristics of the test specimens are
summarized 1n Table 3.2 and Figure 3.4. Flexural tension cracks did
not occur in four of the tests, and in all tests that had cracking
except Arch-7, it first appeared at or above 50 percent of the ultimate
load. In these cases if there were a safety factor against failure of
at least 2.0, then cracking would not occur at service load. In
reality cracks would be less likely to occur due to flexure 1n the
field than shown by these model tests, because creep strains in the
tests were small. When the compression zone of the concrete section
creeps with time as the load is applied slowly, this zone shortens and
the tension stress present on the other side of the section 1S
relieved. Also, if the tension stress is applied over a long period,
the concrete can creep in tension and a larger strain is required to
cause cracking than would have occurred if the tension is applied more
quickly.
There is a definite increase in the tendency toward cracking as the
flexibility ratio becomes smaller as shown in Figure 3.4. This results
from larger deformation and larger moment, which is consistent with the
effect of flexibility ratio discussed above. In Arch-7 where cracking
occurred at 40 percent of the ultimate load, the smallest flexibility
ratio was combined with the rigid block loading that provided the
greatest load concentration. In most of the tests in which cracking
occurred, the width of the crack remained small during a considerable
part of the remaining loading, and started to open significantly only
near failure.
57
1.0
ul ~
0.9
eta.
.
O.8~
Arc
h-5
A
rch
-9
• IT
llJ
m::o
. •
[]]]
] A
rch
-IO
"0
0 0 ...J
0.7
1 C
RA
CK
ING
Q
) -0 0
.6,
• [f
]TI8
E
-A
rch
-2
:::>
~
---I
Arc
h-7
------
NO
-CR
AC
KIN
G
\.J1
00
0'
1 c:
0.4
o
X
u 0 ~
u 0.
3 0
0-
II) ~
lJ....
0.2
'f
- 0 0 0.
1 -0 0:
::
00
t
10
00
f
20
00
3
00
0
12
0
1200
3
65
0
Fle
xib
ility
Rat
io 1
F
FIGU
RE 3
.4
FIRS
T CR
ACKI
NG A
S A
FUN
CTIO
N OF
FL
EXIB
ILIT
Y
RATI
O
, ,-
"
Effect of Reinforcement: Though reinforcement had little influence
on the load capacity of the lining, it had a considerable influence on
its overall behavior. The effects of reinforcement are assessed by
comparing Arch-10 (1.0 percent reinforcement) with the companion test
Arch-9 (unreinforced) and the rest of the arches. The normalized load
to Arch-1 for Arch-9 was 44 kips (196 kN), while that for Arch-10 was
4S kips (200 kN). If all the bars reached their yield stress they
could resist a thrust of 2.5 kips (11 kN) so it is reasonable that this
thrust would result in a small increase in load. By comparing the
general appearance of the cracks for the reinforced specimen with those
for the comparable unreinforced ones, it appears that the reinforcement
serves to distribute the cracks and by so doing keep them finer. When
no reinforcement was present, only one or two cracks appeared 1n the
high moment region near the crown or near the edge of the loaded zone,
and when there were more than one, generally only one of them opened
significantly while the others remained small. However, in Arch-10
with reinforcement, four cracks formed that were approximately evenly
spaced and each of them opened at about the same rate. The cracks at
ultimate load were four times as wide in the unreinforced specimen
(Arch-9) than in the reinforced one (Arch-10). Also near failure
reinforcement held the failed region together for a little more
deflection at nearly constant load.
3.1.2 Circles In Soft Medium
Test Description: The purpose of the circle tests was to
investigate the effects of reinforcement, medium stiffness and joints
between segments on the overall structural behavior of linings 1n a
medium with comparable deformability to soil. Five circular concrete
linings, three monolithic and two segmented, 44 in. (lltwoO rom) in
diameter, 1.0 in. (25 mm) thick and 12 in. (305 mm) long were tested.
Loads were applied through the four center rams 3, 4, 5, 6 as shown in
Figure 3.5. The remaining four rams 1, 2, 7, 8 provided passive
support at the loading surface. A detailed description of the tests is
given 1n Chapter 4 of Volume I and in Sgouros (1982). Since the
primary interest was 1n behavior of the lining, the ground was
59
' .. ~
• I
. ' . . . , .. , ' . ... t': : ",'. of,". ,.... . ', ... '. . .' ., -. ,'. 70 In. ' _ ' . '.' •... '., ~.' (178cm) '.'
' ... '" '., ........ , .... : :' .;/ , . "
.:." .
12in. (305mm)
~'--------11-- Load ing Rams
"'+-+~~r---- Steel Plates
7--- Concrete Stiffeners
Styrofoam -Concrete Fly-Ash Medium
Active Rams; 3,4,5,6 Passive Rams: 1,2,7,8
FIGURE 3.5 TEST SET UP FOR CIRCULAR LININGS
60
represented by a cement-fly-ash-styrofoam bead mix with the requirement
that the deformation should be as large as a soil under the same load.
An additional requirement was that the m1X should have enough strength
to transmit the loads from the loading rams to the lining. Because of
this additional requirement the relative initial stiffness between the
medium and the lining (expressed by the flexibility ratio F) remained
high, as shown in Table 3.3, ranging from 170 for Circle-l to 310 for
Circle-4. However, because of the reduction 1n stiffness of the
cement-fly-ash-styrofoam medium with load, it was estimated that the
flexibility ratio near ultimate was as low as 10 percent of the initial
values.
The circle diameter of 44 in. (1120 mm) as compared to a typical
diameter of a subway tunnel of 20 ft (6 m) corresponds to a model scale
of about 1:5. Applicability of model test results to full scale
tunnels is discussed in detail in Section 2.3 of Volume I. In view of
the limitations imposed by problems with producing and testing an exact
scaled model,
should be done
performed under
extrapolation of the test results to full scale tunnels
with caution. However, since all the tests were
the same conditions, the comparison between various
models is reasonable.
Effects of Reinforcement: The effects of the amount of
circumferential reinforcement on the load carrying capacity of the
lining are examined by comparing Circle-2 with 1.0 percent
reinforcement and Circle-3 with 0.6 percent reinforcement. As shown in
Figure 3.6, the capacity of Circle-3 is higher even though the amount
of reinforcement is lower, because of the slightly higher modulus of
the medium: 40,000 PS1 (275.6 MPa) for Citcle-3 and 35,000 psi (241.1
MPa) for Circle-2. Thus, the load carrying capacity of the lining is
more sensitive to variation of the modulus of the medium than the
amount of reinforcement present. Reinforcement does little to improve
the load carrying capacity of the lining, because the reinforcement is
most effective 1D tension, and for the flexibility ratio range
considered, most of the lining section is in compression.
61
I
TABLE 3.3 SUMMARY OF MODEL CIRCULAR LINING TEST RESULTS
Circle-l Circle-2 ----------- --------------
Type of Lining Monolithic Monolithic
Amount of Reinforcement (%) 0 1
Initial Equivalent Elastic Modulus of Medium 25,000 35,000 psi (MPa) (172.2) (241. 1)
Compressive Strength of Lining Concrete 2,760 2,560 f' c' psi (MPa) (19) (17.6)
Peak Load 48.4 73 Kips, (KN) (215) (325)
Change in At 50% of Peak Load 0.36 0.43 Diameter
~ (%) At 100% of Peak Load 1. 27 1. 20
First Flexural Cracking Load, % of peak Load 42 36
Crown Crack Size at 90% of Peak Load, 0.01 0.001 -in. (mm) (0.3) (0.03)
Failure Section, Degrees from Crown SO-Left 70-Left
Initial Flexibility Ratio, F 170 250
Estimated Flexibility Ratio, F at 90% of Peak Load 17 25
Thrust Ratio (T)To) at Failure 0.5lc 0.56c
~onolithic flexibility, joints not taken into account.
bMeasured.
CEstimated.
62
Circle-3 Circle-4 ------------ ------------Monolithic Segmented
0.6 1
40,000 45,000 (275.6) (310)
2,240 2,720 (15.4) (18.7)
90 77 (400) (342)
0.25 0.41
1. 30 1. 20
32 91
0.004 --(0.1)
65-Right 60-Right
300 3l0a
30 31
0.62b 0.56b
Circle-5 ------------
Segmented
1
35,000 (241.1)
2,280 (15.7)
70 (311)
0.36
1.86
93
--
l20-Lef t
260a
26
0.40c
,;.,~'
CRO
WN
D
EFL
EC
TIO
N
(MM
) 5
15
2
5
55
0
10
2
0
50
4
0
10
0
q0
C
ircl
e-3
40
0
80
5
50
70
C
/) 5
00
a.
z
.... 6
0
. Ci
rc 1
e-2
~
~
25
0
Cl
0 <I
5
0
a:
0 0
0\
J 2
00
-l
w
...J
40
-l a:
<I
t-t-
15
0
0 0
t-t-
50
C
ircl
e-l
r00
2
0 -
I-r
/ I/
Hd .-
Fi r
s t
Cra
ck i
ng
10
II
1/
50
o "
I 1
0
0.
.4
.8
1.2
1
.6
.2
. 6
1 •
1 .4
C
RO
VN
D
EFL
EC
TIO
N
(IN
)
FIGU
RE 3
.6
LOAD
-DEF
LECT
ION
CURV
ES
FOR
THE
MON
OLIT
HIC
LINI
NGS
Flexure related cracks appeared at the crown and at the springline
in all three monolithic lining tests. Overall additional cracking was
more severe in Circle-1 (Figure 3.7) than 1n Circles-2 and Circle-3
consistent with its low modulus and lqck of reinforcement.
Circle-2 to Circle-3 1n Figures 3.8 and 3.9, it is observed
Comparing
that less
cracks are present 1n Circle-3 even though the reinforcement ratio is
smaller, because of the larger medium modulus. The beneficial effects
of reinforcement are observed in Figure 3.10, where the width of the
crown cracks for the three monolithic linings is compared. Though
initial crack width is less than 0.004 1n. (0.1 mm) in all three
linings, the crack opens with additional load in the unreinforced
specimen but maintains the same width in the reinforced ones.
90 percent of the peak load the crack 1S three times wider
unreinforced specimen.
At about
1n the
The time of appearance of first cracks is affected very little by
the amount of reinforcement,
strain of the concrete. First
s1nce it is a function of the cracking
cracks appeared at about 32 to 42
percent of the peak load in all three monolithic tests as shown 1n
Figure 3.10. First cracks in the two segmented linings Circle-4 and
Circle-5 appeared at 91 and 93 percent of the peak load. However, this
was the result of the presence of the joints rather than of the
reinforcement in the lining. The conclusions regarding reinforcement
concern short term load-related cracking. The effects of reinforcement
on shrinkage or temperature related cracking or the long-term behavior of
the lining cannot be obtained from these relatively short term tests.
If the loads were applied slowly and the concrete could creep during
the load application, the cracks would occur at higher loads and would
not open as much.
Effects of Medium Stiffness: The stiffness of the medium is the
most important parameter in determining the load carrying capacity of
the lining. The higher the modulus of the medium, the higher the
capacity of the lining as shown in Table 3.3 for Circles-I, 2 and 3. A
stiffer medium decreases the lining deformation and thus the moments,
so the thrust ratio T IT increases; the higher the modulus of the u 0
64
Top View
I) \17
Side View
OO(Crown)
10
17
Top View
C 7 \
Side View
0° (Crown)
1---__ ---"20
14
Top View
Side View
r
Failure Zone
180° (Invert)
15
o
FIGURE 3.7 CRACKING PATTERN OF CIRCLE-l
\ \ ,,:1 I
Failure Zone
9 9
FIGURE 3.8 CRACKING PATTERN OF CIRCLE-2
7 ( I
Failure Zane
1800 (lnvert)
---18 16 18
7
FIGURE 3.9 CRACKING PATTERN OF CIRCLE-3
65
CRACK SIZE (IN) .004 .012
.008 .016 0. 100 ~~----~~--------+--------1--------cd======~
80
70
o 60 a: o ~
~ a: 50 w a.. u.. o x 40
30
20
10
Circle-3 -Circle-2 +--
Circle-l
Circle-l: Unreinforced
Circle-2: Reinforced -1 Percent
Circle-3: Rei nforced -0.6 Percent
o ~ __ ~ __ ~~--4----+----+---~----~--4---~--~ 0. .1 .2 .3 .4 .5
.05 .15 .25 .35 .45
CRACK SIZE (MM)
FIGURE 3.10 EFFECT OF AMOUNT OF REINFORCEMENT ON CROWN CRACK SIZE OF MONOLITHIC LININGS
66
,~ ,
medium, and thus the flexibility ratio, the higher the thrust ratio.
Thus, preserving the integrity of the tunneled ground by a coordinated
excavation and initial support sequence will not only reduce the loads
that reach the final concrete lining, but it will also enhance its
ability to support these loads if and when they occur. Furthermore,
effective backpacking of the void between the concrete lining
(cast-in-place or segmented) and the inside face of the tunneled
opening 1S very important 1n ensuring the much needed passive
resistance and proper redistribution of external loads and internal
forces.
Effects of Joints (Monolithic vs Segmented Linings): Some
indication of the effects of the joints on lining behavior may be
obtained by comparing the load-deflection curves 1n Figure 3.11 of
Circle-2 (monolithic) and Circle-5 (segmented) with the same medium
modulus; the capacities of the linings are comparable, with the
segmented lining exhibiting a slightly lower capacity due to the
reduction of the cross-section at the joints.
The joints between the segments act as pre-formed cracks and thus
by rotating during loading they decrease the moment in the lining.
This in turn results in fewer cracks as shown in Figures 3.12 and 3.13.
Furthermore, first cracks appeared at over 90 percent of the peak load
for the segmented linings and their S1ze was only about one-half of
those 1n the monolithic linings. The effect of joints in reducing
moments 1n the lining is magnified as the stiffness of the medium
becomes lower, S1nce the joint rotations 1ncrease, as observed by
comparing the crack patterns of Circles-4 and 5. The changes in lining
diameter are very close for the monolithic and segmented linings. In
full scale linings the difference between segmented and monolithic may
not be as dramatic as in these tests, because segmented linings are
normally constructed with offset longitudinal joints between adjacent
rings, and therefore the actual stiffness is between the values for a
monolithic lining and that for a single ring of segments. The range of
AD/D at 50 percent of the load is from 0.25 to 0.43 percent and at the
peak load from 1.20 to 1.27 percent (Circle-5 excluded).
67
CRO
WN
D
EF
LE
CT
ION
CM
M)
2.5
7
.5
12
.5
17
.5
22
.5
e;,.
5.
le;,.
1
5.
2e;,.
2
5.
Se;,
I I
I I
I Ir55~
rlI-
Ol
7~ I
Cir
cle-
2 (M
onol
ithi
c)
t 3e;,e;,
se;,
(Seg
men
ted)
+
25e;
,
~ 6C
t /
I z ).
/.
20
0
0
~ 4C 1
t 15C
a:
0 ...J
C'
...J
00
a:
I-
-a:
3e;,
0
I--
I--
a I--
10
0
2e;,
1e;,
50
e;, 0
0.
.2
.4
.S
• S
1 •
• 1
.3
.5
.7
.q
CRO
WN
D
EFL
EC
TIO
N
(IN
)
FIGU
RE 3
. 11
COM
PARI
SON
OF
LOAD
-DEF
LECT
ION
CURV
E OF
MON
OLIT
HIC
AND
SEGM
ENTE
D M
ODEL
S CI
RCLE
-2 A
ND C
IRCL
E-5
-.'~ ,"
-.'
,
Top View
.Side View [,Olur n
OO(Crown) 9?0 1800( Invert) 270° 36 1
F·, e Zo e
12 II -~
) 13 13
12
~ FIGURE 3.12 CRACKING PATTERN OF CIRCLE-4
Top View
( :\ fl6 (
.Side View omt oea Ion f Qlure one
OO(Crown) 90° 1800( Invert) 270° 36
I I
L t FI Z
~ 15~~ ~
~ I~~ 15
N ~ " \
16
\' 15
FIGURE 3.13 CRACKING PATTERN OF CIRCLE-5
69
Failure Modes: Failure of the linings occurred between 60 and 80
deg from the crown with the exception of Circle-5 where it occurred 120
deg from the crown at a joint. All the model lining failures resulted
from crushing of concrete that began at the inside surface in some
small region and with additional load it spread both through the depth
of the section and longitudinally 1n the lining. The initiation of
crushing was generally accompanied by cracking parallel to the
direction of compressive stress (circumferential in the lining) typical
of compression failures in concrete and to be expected 1n a specimen
only one inch thick. Crushing began on one side of a joint in both
segmented lining models. Circle-4 failed on both sides of the joint
(Figure 3.12) while 1n Circle-5 crushing occurred only on one side
(Figure 3.13). The joints constitute a weak section because of the
reduced cross-section and the lack of reinforcement. There was no
indication of high shear stress contributing to the initiation of
failure or contributing to the spread of failed concrete after
initiation (as might be indicated by a radial offset each side of the
failure region if shear were a contributing factor).
3.2 ANALYTIC STUDIES
3.2.1 Parameter Study of Arches
Parameter studies were performed on semicircular arches with
loosening loads uS1ng the beam-spring model to show the effect of
flexibility ratio F, radius to thickness ratio R/t, ratio of tangential
to radial stiffness of the springs K /K and load shape on a typical t r
full scale station configuration. The model used is shown 1n Figure
3.14; radial springs in tension were inactive but all the tangential
springs remained active. Nonlinear behavior of the lining was allowed
due to material behavior and geometry change. A typical stress-strain
curve for concrete was used. Radii of 20, 25 and 30 ft (6, 7.6, 9 m),
thicknesses of 10, 12 and 15 in. (250, 300, 380 mm) and a concrete
compressive strength of 4000 psi (27.6 MPa) were investigated. Most of
the study was performed for a 12 in. (300 mm) thick arch of 25 ft (7.6
m) radius with a uniform load across the full arch and several problems
70
.,
.. ~.
12" (305 ) mm I' 'j
12" I~ (305mm) ~
f' ::: 4.0 ksi (28 t~Pa) c
T ::: 576 kips(2562 kM) o
(10-15")
(254-381 mm)
12" (305 mm) I' ·1
lEI]..:l:.. 2/1 ...•. ). .....
A " T
' . .,< : .. :. .1.. 211 T
(51 mm)
(31 mm)
f I ::: 4 0 k s i (2 8 ~W a) c . f = 40 ksi (276 r'lPa)
A~ ::: A~ ::: 0.48 in 2(3.0 mm2)
T ::: 518.0 kips (2036 k~) o ? 2
A = A' ::: 0 78 in-(503 mm ) s s . To ::: 782.4 kips (3480 kN)
of Elements::: 18
of Joints::: 37 No. of Radial Springs::: 19 No. of Tang. Springs::: 17
No. of Base Springs::: 1 No. of Equations::: 73
FIGURE 3. 14 FINITE ELEMENT MODEL FOR PARAMETER STUDIES OF ARCHES
71
were worked with a symmetrical triangular load over the center 60 deg
portion of the arch; to investigate further the effect of load shape
one problem with uniform load over the 60 deg portion and one with
uniform load over the right one-half of the arch were worked. The
radius and thickness were varied in some problems to study their
effects, while keeping the K /K ratio constant at 0.25. These t r
solutions were for arch sections with one-half percent reinforcement 1n
each face, and then a ser1es of solutions were obtained for an
unreinforced section for the full range of flexibility ratios and one
value of tangential shear stiffness (K /K = 25 percent). As discussed t r
1n Section 3.1.1 the flexibility ratio F is used as a convenient
measure of relative lining-medium properties even though it is derived
for circular linings.
In Figure 3.15 the moment-thrust paths for the most critical
section in the lining are shown on the interaction diagram for a
flexibility ratio of 3500, lining thickness of 12 in. (300 rom), radius
of 25 ft (7.6 m) and various load shapes. The symmetrical uniform load
that covers the full span induces the smallest moment and therefore the
largest thrust before failure. The triangular load over the central 60
deg arc induces the largest moment 'at the crown and consequently the
smallest thrust and load at failure. The other two loading conditions
fall between these two extremes with the unsymmetrical uniform load
(over one half the arch) causing very nearly as much moment as the
triangular load. However, reference to the geologic conditions that
tend to cause various loads indicate maximum loads that can occur; if
joint sets are assumed to occur in both directions relative to the
vertical, a triangular wedge of rock depicted by the triangular load
could occur about one radius wide at the base and have a height of
about one-third diameter. The total weight per foot of tunnel for 50
ft (15.2 m) diameter would then be 31.3 kips (139 kN), if the rock is
assumed to weigh 150 lb/ft3 (2400 kg/m
3). The total load at failure
from the analysis was 277 kips (1230 kN) so there is a safety factor of
8.8 against failure. If, however, the uniform load completely across
the lining is assumed to represent a vertical depth of rock of one
diameter, the total weight is 375 kips (1670 kN) per foot of tunnel.
72
>'. ",
MOMENT (KN-M)
0 20 40 60 80 100 120 700.
Reinforcement 0.5% 650. R/t = 25 2800 600. F = 3500
Kt/Kr = 25~~
550. 2400
500.
450. 2000
en 400. Loading: Uniform Sym. , Z
• ~ a... t -::.L : 1600 350. Uniform Over 60°
I ;l- t-
to- Sym. I (f) (/)
I" 3---. __ ............ :::J
::;:l ..-0.:: 300. # O!: ::r:: ,../ :r: to- 1200 t-
250. -' ,/ 200. /
,'-" 800 ,,'
15H. ","
*/,,~
,," Triangular " 100. 400
50.
0. 0 0. 200. 400. 600. 80H. 1000.
MOMENT (IN-KIPS) FIGURE 3.15 MOMENT-THRUST PATHS FOR CRITICAL SECTION FOR VARIOUS
LOAD SHAPES
73
At failure the load was 992 kips (4420 kN) so the safety factor is 2.6
for this loading. This comparison shows that the triangular load is
less critical though it produces more moment in the lining at a given
thrust and thus results in failure at a smaller thrust and smaller
total load. The safety factors obtained for partial uniform
symmetrical and partial uniform unsymmetrical loadings were 5.0 and
4.8, respectively, which are still higher than the safety factors
obtained for the uniform load across the arch.
Arbitrary dimensions of rock blocks have been assumed in the
comparison above, and in reality they would depend on actual field
conditions, but they are reasonable and show that the condition
providing the greatest total load tends to be most severe. For all the
symmetrical loadings the critical section occurred at the crown, and
for the unsymmetrical one it occurred on the loaded side of the crown.
These solutions were obtained for a lining thickness of only 12 1n.
(300 rom) which may be considered as about the lower limit of minimum
constructable thickness of linings for such large openings. If the
uniform rock load is not expected to be greater than one diameter above
the crown, it may be concluded from the above analysis that the minimum
constructable lining is adequate since the factor of safety for such
loading is shown to be more than 2.
The effect of flexibility ratio F on total load on the lining for
two loading shapes are shown 1n Figure 3.16 where the effect of
K IK is also apparent. The vertical axis is changed to T IT in t r u 0
Figure 3.17, where the curve shapes are very similar because the total
load and the ratio T IT are almost proportional for a given load u 0
shape. This proportionality is shown in Figure 3.18 and depends very
little on the K IK ratio. With this t r
considered the variation of load with
in mind, Figure 3.17 can be
F for different K IK ratios and t r
load shapes. The curve is not as flat as that obtained from the model
tests, but shows a definite decrease in sensitivity to F at larger
values; in this range determining accurate values of rock and lining
stiffness for calculating F are not as critical as it is in the low
range where the curve is steep.
74
1200. Reinforcement o. 5c~ R/t = 25 Kt/Kr = 4900
25:; 1000.
12.5% 4200
00/ ,0
800. CJ) 3500 a.. z ..... Uniformly Distributed Load Symm. ~ ~
2800 0 0 a: a: 600. 0 0 ...J ...J
...J ...J 2100 a: a: l-I- 0 0 400. Triangular Load Synm. l-t-
1400 2 5~~
200. 700
0. ~ ____ ~ ____ -+ ______ ~ ____ ~ ______ ----__ ------40
0. 2000. 4000. 6000. 3000. 5000. 7000.
FLEXIBILITY RATIO
FIGURE 3.16 TOTAL LOAD VS FLEXIBILITY RATIO FOR ARCHES
75
1.
.9
.8
~
.7
1-0
.....
... :::l
t-.6
0 .....
I-.5
a:
O
l ....
... 0
-t;
.4
:J
Ol
I t-.3
.2
• 1
0.
0.
Rei
nfor
cem
ent
0.5%
. 5
K /
K
= 2
5°;
____
_ )(
t r
----
----
---
I R
/ t
=
16
____
_ -*----
--------
--------
;-;:1--
---------
!>
I "
.,.-
, ... .,_
... -'
~-
----
----
25
1 2.
5 j;
____
_ ---
---*
>t~--~-
____
-~
Tri
anC
ju1a
r Lo
ad
(all
o
thers
u
nif
orm
lo
ad)
1.
2.
3.
4.
5.
1.5
2
.5
3.5
4
.5
5.5
F
LE
XIB
ILIT
Y
RA
TIO
X
1
00
0
FIG
URE
T~17
THRU
ST
RATI
O
VS
FLEX
IBIL
ITY
RA
TIO
FO
R AR
CHES
"',
' -'
"
f,'
6 .
TOTAL LOAD (KN)
800 2400 4000 0 1600 32.00 4800
1. 1. Reinforcement 0.5%
R/t = 25 .9 .9
UDL Sym.
.8 I K K = t r .8 25;;
.7 .7
0 Triangular Load I-.......... :::::l.6 .6 I-
0 ....... I-a: 0£:
I-Cf) ::J 0£: :c I-
Kt/Kr = 25;~
.5 .5
.4 .4
.3 ...,.
.0
.2 .2
. 1 • 1 UDL = Uniformly Distributed Load
~~---+--~--~--+---r-~~-+---+--~--~--+0. 0. 0. 2.00. 4.00. 600. 80.0. 1.0.0.0. 12.0.0.
TOTAL LOAD (KIPS}
FIGURE 3.18 THRUST RATIO VS TOTAL ULTIMATE LOAD FOR ARCHES
77
0
The influence of F on the ratio of change ~n radius to radius
(~R/R) at failure in the direction of loading is shown in Figure 3.19.
Linings ~n a soft medium (or low F) show considerably higher
deformation, which decreases sharply as the flexibility ratio becomes
larger. The deformations are also considerably higher for the case of
no tangential shear than for the other values of K /K ratios. t r
Figure 3.20 shows the effect of tangential shear stiffness K /K on t r
the moment-thrust behavior for the two extreme values of F equal to
6000 and 285, and the full uniform loading. For both values of F the
curves are shown for K /K of 0, 0.125 and 0.25. For each value of F t r
the radial spring stiffness rema~ns the same, and the tangential
stiffness changes. The moment-thrust paths reach the interaction
diagram above the balance point for the larger F and below it for the
smaller. For both values of F there is an increase in total load with
K /K increase, and the tangential stress between the medium and lining t r
can have a substantial effect on peak loads. Also, the absolute value
of increase in load due to tangential shear increase is larger for the
larger flexibility ratio, but the relative increase is smaller. The
increase in load for an increment of K /K from 0 to 12.5 percent ~s t r
greater than that for 12.5 to 25 percent. This effect is more
pronounced for large flexibility ratio, as shown also in Figure 3.21
where T /T ~s plotted against K /K . u 0 t r
A lining with the same dimensions and concrete stress-strain curve
was investigated without reinforcement. The ultimate pure thrust
capacity (T ) is reduced to 576 kips (2560 kN). The moment-thrust o
envelope for the unreinforced lining is also smaller than the
reinforced one ~n Figure 3.22 where the moment-thrust paths for
different flexibility ratios and for K /K = 0.25 and R/t = 25 are t r
shown. The moment in the unreinforced lining is slightly smaller at a
given thrust than that in the reinforced one because removal of the
reinforcement decreased the lining stiffness and therefore increased F,
and an increase ~n F causes a decrease in moment. However, this
difference in the initial moment-thrust ratio ~s small. As the
moment-thrust paths approach their respective failure envelopes, the
78
Ol "- Ol
.....
., 1.
0
·02
.01
8
.01
6
.01
4
.01
2
. 01
.00
8
.00
6
.00
4
.00
2
0.
0.
1.
. 5
Re
info
rce
ille
nt
0.5%
R/t
=
25
Kt/
Kr
= O
~0
12.5
<J:
-----~ _
_ ~25~
%~
I *
2.
3.
4.
5.
6 .
1.5
2
.5
3.5
4
.5
5.5
FL
EX
IBIL
ITY
R
ATI
O
X 1
000
FIGU
RE 3
.19
VARI
ATIO
N OF
RAD
IUS
CHAN
GE
RATI
O AT
FAI
LURE
WIT
H FL
EXIB
ILIT
Y RA
TIO
MOMENT (KN-M)
0 20 40 60 80 100 120 700.
Reinforcement 0.5% ---- F = 6000
650. R/t = 25 -F= 285 2800 600.
550. 2400
500. \ 450. + 2000
! ~
! I I :z
CfJ 400. ! 0-
, ~
Kt/K = 25:{,! ,
........ / :::.L r / I 1600
350. , ;' l-I
/ ,/ I-- / . (J) I
CfJ ,1/ .
::;:, ,I ,,,,,/ ::J
o.!: 300. ,,/ OL :::c l' 12 5'" , ,,2" ::r: I--
• ,? / ,
//" 12.00 l-
I
250. /It ,"
,/ 25°1, / ,/' "".;I'"
200. / 0°/ ",
/,' /0 ","
l ,1" ",/ 800 "
;,' ",'
150. I ;' / /" ,/ ,/ "" / / ,...2
. -
100. ,.
/ / ,,/ 400 ,~ • ~> /;' ", , / ,"
,.
/ / "."
50. /,' ,/ , , .. ,I" ,,,,,,,/
0. ~,~;,,'
0. 200. 400. 600. 800. 1000.
MOMENT (IN-KIPS)
FIGURE 3.20 EFFECT OF Kt/Kr ON MOMENT-THRUST PATHS FOR ARCHES
80
1 •
. 9 .8
F =
60
00
1-0
.7
"-:J
I-6
~.
3500
....
....
.•.•
••..
•.••
.•••
••••
••••
.•..
••••
..••
• lI .•
.•.•
•.••
...•
.•.•
.•..
•...
....
....
.•..
...•
...•
....
..• l<
... ......
........
.. 11
00
0 .....
I-.5
a:
Q£
I-.4
(f
j
500
••••
••••
-...
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
...
4
285
00
.....
:J
Q£
I
.5
I-
.2
• 1
0.
0.
.04
.0
8
. 1
2
.16
.2
.2
4
.28
.0
2
.06
•
1 .
14
.
18
.2
2
.26
Kt/
Kr
FIGU
RE 3
.21
EFFE
CT O
F K
t/Kr
ON T
HRUS
T RA
TIO
FOR
ARCH
ES
: ;.
"<
-I'
. ~ ;
-
MOMENT (KN-M)
0 20 40 60 80 100 120 700.
< , ,-"'~ ",
Kt/Kr = 2 5~~ - Reinforced
650. R/t = 25 ---- Unreinforced
2800 600.
Unreinforced Envelope
550. 2400
500.
450. 2000 z ,
(fJ 400. a.. ~ ,
-~ 1600 350. l-
t-- (f) (fJ :::J ::;:, 0-::: 300. OL ::::c :r: t-- 120fl I-
250.
200. 800 0
.-
150.
100. 4fjfl
50.
0. 0
0. 200. 400. 600. 800. 1000. '"
MOMENT (IN-KIPS)
FIGURE 3.22 EFFECT OF REINFORCEMENT ON MOMENT-THRUST PATHS FOR ARCHES
;' ,- .
82
ultimate thrust (T ) and correspondingly the maX1mum load for the u
reinforced lining is higher than the unreinforced lining because the
envelope 1S higher. However, the ratio T /T increases for the u 0
unreinforced lining by a small amount as shown in Figure 3.23. This
difference in T /T for the two cases would be reduced, however, if the u 0
actual values of F were computed for the reinforced section based on
the transformed section.
Solutions were obtained for values of radius to thickness ratio of
16, 25 and 36 based on radii from 20 to 30 ft (6 to 9 m) and
thicknesses from 10 to 15 in. (250 to 380 rom). The effect of the R/t
ratio was studied by keeping the K /K at 25 percent the reinforcement t r '
ratio at 0.5 percent in each face and varying the dimensions. Figure
3.17 shows that as the R/t ratio increases, thrust ratio decreases when
F and K /K are kept constant. The variation of R/t from 16 to 25 and t r
from 25 to 36 renders similar changes in thrust ratio. Thus linear
interpolation is possible for intermediate values of R/t and the same F
and K /K . t r
In summary, the parameter study indicates that a uniform loading
across the entire arch has the smallest safety factor against collapse
for the particular conditions investigated. Though a triangular and
unsymmetrical uniform loading provide lower values of T /T they still u 0'
have larger safety factors for the rock block dimensions selected.
This study was not broad in its coverage of parameters, however, and
the unsymmetrical loadings should be investigated at a particular site.
The studies also show that strength increases with F though there
1S a definite flattening of the curve in Figure 3.17 above F = 1000,
indicating less sensitivity to the calculation of F. An increase 1n
shear stress between the lining and medium results in a definite
increase in strength of the lining. Therefore, it is essential to
include the shear stress 1n the analysis of the lining to obtain a
realistic prediction of strength, especially for larger values of F,
but the strength 1S not highly sensitive to K /K as shown by the t r
flatness of the curves in Figure 3.21.
83
1.
.9
.8
Unr
ei n
forc
ed ~ __
_ ---
----
----
----
-.. ------~:::::--
------
.7
0 t-- "-
:::l
_----~
"-
Rei
nfo
rced
(0
.5%
)
----
----
----
----
---
-----I-
.6
0 ~
.5
a: ~
00
~
I-.4
(f
) ::
l ~
I t--.3
.2
. 1
Kt/
Kr
25%
0.
0.
1.
2.
3.
4.
5.
6.
.5
1 .5
2
.5
3.5
4
.5
5.5
F
LE
XIB
ILIT
Y
RA
TIO
X
100
0
FIGU
RE 3
.23
EFFE
CT O
F RE
INFO
RCEM
ENT
ON T
HRUS
T RA
TIO
FOR
ARCH
ES
.. ~:
Reinforcement increases the absolute strength approximately as
would be indicated by an analysis of the section, but both the model
tests and parameter studies show that the effect on the T IT ratio is u 0
very small. That is, T and T increase in the same ratio, so the u 0
reinforcement has little effect as might result from changing the
relative stiffness of the lining and medium.
3.2.2 Parameter Studies of Circular Linings in Soft Ground
Description of the Program: An existing nonlinear finite element
program that used springs to represent the medium was modified to
provide a better representation of soft-ground conditions. The
existing program used a special three-node beam element to represent
the lining that can model reinforced or unreinforced concrete sections
with nonlinear stress-strain properties of the concrete and
reinforcement. This spring representation for the medium does not
properly represent all the interaction components that occur when a
lining is placed in a continuous homogeneous medium because the load
must be applied directly to the lining, and it does not account for
arching of loads around the lining. Hence for better representation of
the medium as well as to be able to handle various other loading
conditions, the program was modified by incorporating two dimensional
8-node quadratic isoparametric elements that will represent the medium
as a continuum. Also a special type of interface element was devised
to represent slip at the ground-lining interface.
For efficient handling of large problems, the solution method was
modified to use a multiple-level substructuring scheme, so the medium
could be divided into a number of linear substructures and static
condensation performed for each one to compute condensed stiffness
matrices and equivalent load vectors. This process of substructuring
could be continued sequentially at subsequent higher levels reducing
the size of the total structural stiffness matrix for the medium to a
size that can be handled by the computer. This scheme also allows the
use of a particular substructure representation of the medium
repeatedly in several problems with the same medium geometry but
85
different elastic moduli without recomputing the stiffness matrix. An
interface element was devised, (Saha, 1982), that is placed at each
node and oriented tangent to the lining curvature at the joint. This
element has been modeled with two relative displacement degrees of
freedom, one in the normal and the other in the tangential direction,
and zero stiffness 1S assigned to the elements when separation occurs.
For the purpose of parametric studies, only nondilatant joint
properties were given. Very high normal stiffness is assigned to model
contact and a Mohr-Coulomb criteria 1S used to model elasto-plastic
tangential shear deformation at the interface.
Description of the Study: In the analysis of linings 1n soft
ground the greatest uncertainty lies in the way the ground loads reach
the lining. Some of the commonly used loading conditions are described
in Section 1.2.3 and are termed i) overpressure loading,
ii) excavation loading, and iii) gravity loading. In the parametric
studies the lining behavior under these loading conditions were studied
and compared in view of the degree of their severity at both ultimate
and first cracking levels. Linear closed form solutions for the first
two types of loading were also solved for a wide range of parameters
and compared with the corresponding linear and nonlinear finite element
solutions using the modified program. The comparison of the linear
solutions also served to verify the program, while that with the
nonlinear solution resulted in an evaluation of the effect of
nonlinearity of the problem in terms of redistribution of internal
forces and additional strength over that predicted by linear analyses.
Loading conditions consisting of water pressure or removal of internal
air pressure were also investigated. Other variables considered were
the interface properties such as cohesion c and internal frictional
angle $, shear
reinforcement.
modulus G, coefficient of earth pressure K and 0'
The effects of these parameters on the behavior of the
lining are presented briefly 1n the following sections.
Effect of Interface Properties: The slip condition at the
ground-lining interface is controlled by three parameters defining the
material properties for the interface element: cohesion c, angle of
86
internal friction ~, and shear modulus of the medium G. Variation of
cohesion within reasonable limits (0 to 5 psi) did not have an
appreciable effect on the solutions; there 1S very little difference 1n
the crown deflection, tangential and radial pressure distribution, or
moment and thrust distributions as shown in Figures 3.24 and 3.25. For
this reason a small value of c has been used in most of the problems
for parametric studies as it facilitates obtaining convergence.
Variation of the angle of internal friction did show an appreciable
effect; with high ~ (= 45 deg), the shear strength of the interface
increases and larger tangential shear stresses result, which reduces
the thrust at the crown and invert. The thrust distribution for a
small ~ remains fairly uniform since the tangential shear strength
remains low at the interface, which leads to a condition near full slip
as most of the interface elements become plastic in shear at this load
level. On the other hand for higher ~, a partial slip or nearly no
slip situation arises. Moments are larger for low ~ and S1nce crown
thrust is lower, the critical section 1S at the crown for low
flexibility ratios (F < 7). However, the total load does not differ
appreciably between full slip and partial slip or no slip conditions as
shown in Figure 3.26. On the other hand for higher flexibility ratios
(F > 7), the thrust at the springline becomes so much higher for the
partial slip case that the failure section changes to the springline
and failure occurs at a lower load than in the full slip case. Thus
the load capacity is lower for the partial or no slip case and the
difference increases with the increase in flexibility ratio (Figure
3.26). The same trend occurs for excavation loading also.
Effect of Water Pressure: For studying the effect of water
pressure on the lining behavior and on the total load capacity, uniform
all-around water pressure was added to the ground loads in the
incremental solution process until an estimated service load was
reached and then the incremental ground loading was continued. The
moment-thrust paths shown in Figure 3.27 for three different water
pressures (0, 2.5 Y D and 5 Y D where Y is the unit weight of water w w w
and D is the lining diameter), clearly indicates the benefi effects of
water pressure. Since a uniform pressure does not induce moment, only
87
TANGENTIAL STRESS ( PSI ) 100
RADIAL PRESSURE ( PSI )
50 .-=--=-=--_--s-.._----_
INTERFACE ELEMENT PROPERTIES C> 0 ¢= 45j c = 0 PSI 6.-----~ cP= 45j c
,... = ) PSI
1l- . - .--{) ¢= 10; c = 5 PSI
TOTAL LOAD = 80,0 KIPS
50
100
50
o
\t \
\ '
\
~ \ '
FIGURE 3.24 EFFECT OF ~ AND c ON RADIAL PRESSURE AND TANGENTIAL STRESS
88
'II' ..
30
I
1.1
:
I I '. I \
AXIAL THRUST (KIPS)
/
¢
/
10
--- r1or1ENT (KIP-IN)
INTERFACE ELEMENT PROPERTIES Q 0 ¢ = 45~ c = 0.0 PS I ~-----c:. r:j)= 45; c = 5.0 PSI tr-._.-u cp = 10; c = 5.0 PSI
TOTAL LOAD = 80.0 KIPS
'$_
-~-
2.0 1.0 -1.0 -2.0
FIGURE 3.25 EFFECT OF ~ AND c ON AXIAL THRUST AND MOMENT
89
u.. !IJ ~
S0
50
.... 40 Q <I o .J
W ... <I 50 1: ..... ... ...J ::J
20
10
Nonlinear Analysis Overpressure Loadinq
= 0.4 'Ym
Full SliP"/
/ "
Partial slip <p = 200
; c= 3.5 psi
(interface element properties)
0.5% reinforcement each face R = 118 in. (3.0 m) t = 10 in. (254 mm) f 1= 4000 ps i (27.6 MPa)
c
10 2" 4" 5 15 25 55
FL~XIBILITY RATIO, F
FIGURE 3.26 EFFECT OF SLIPPAGE ON ULTIMATE LOAD
90
45
2700
2400
18"0 ~ a ..l
w 1500 I
a: 1: M
I-
1200 5
MOMENT (KN-M)
" 1" 2~ 3" 4" 5" 6" 5"". 22~"
---- Springline 45". -_ Crown 2~""
Unreinforced Deep Tunnel
4"". Full Slip 18"~ F = 7.8
15", Water Pressure/yw = 5.00
Service load-r_,·-.....-::::::;::;.-'7'o
5a. No water pressure 2~"
a. ~~+---~--~--~~~~~~~~--~--~~~--~ " " . " 1 "a . 2~" . 0"" . 4"a. 5"" . 6"" .
MOMENT (IN-KIPS) ' ...
FIGURE 3.27 EFFECT OF WATER PRESSURE ON MOMENT-THRUST PATHS
91
the thrust 1S increased with the increase 1n water pressure until
service load. This reduces the slope of the moment-thrust paths
(eccentricity) which helps 1n avoiding or reducing cracks in the
lining. Table 3.4 summarizes the effect of water pressure on various
other parameters and indicates that although the thrust ratio increases
with the increase in water pressure, total ultimate vertical load 1S
not changed appreciably.
Effect of Loading Conditions: Three different loading conditions
were investigated to determine their effect on the lining behavior
under different conditions of slip for several flexibility ratios. The
overpressure loading condition was modeled by applying uniform pressure
at the top surface of the initially unstressed medium with the lining
1n place. The excavation loading condition, explained in Section
1.2.3, was obtained by applying the shape of the in situ stresses at
the interface
proportionally.
on the medium nodes and increasing
The excavation and overpressure loading
the magnitude
were applied
to a lining in a linear medium with joint elements between the lining
and medium having properties corresponding
friction ~ of 20 deg and cohesion of
to an angle of internal
3.5 psi (0.024 MPa). The
beam-spring model was used for the gravity loading cases. A wide range
of variables and loadings were investigated for these loadings
conditions for a 10 in. (254 mm) thick lining with 118 in. (3.0 m)
radius and 0.5 percent reinforcement in each face. For selected cases
for the gravity loadings, the problems were also run for the same
lining but without reinforcement. The compressive strength of the
concrete was 4000 psi (27.6 MPa).
The difference in capacity between the overpressure loading and
excavation loading increases with flexibility ratio up to a flexibility
ratio of about 20, and then remains fairly constant as shown in Figure
3.28. The overpressure loading will generally lead to an overly
conservative design and the excavation loading is more suitable.
Another important parameter 1n connection with the excavation
loading is the coefficient of earth pressure K For very soft soil o
92
1.0
Lv
TABL
E 3
.4
EFFE
CT
OF
t'JAT
ER
PRES
SURE
ON
VA
RIO
US
PARA
t1ET
ERS
To
tal
Ult
imate
F
acto
r ID
at
H 1
flD
E:
L
oad
of
T
IT
cr
sp
w
ksf
S
afe
ty
u 0
Ult
imate
at
Ult
imate
o.
22
.8
5.3
6
.41
9
2.0
.0
04
8
2.5
D
21
. 4
5.0
2
.44
9
1. 8
79
.00
45
-' r-'--
-,-,----
-----_
._---
---_
.----~-.~-----~-
-5
D
20
.0
Eff
ect
of
Not
In
cre
ase
in
A
pp
recia
ble
W
ater
P
ro
Cha
nge
1 H
is
d
efi
ned
as
bel
ow
: w
4.6
9
.47
8
1.
916
.00
45
Not
In
cre
ase
s N
ot
---
Ap
pre
cia
ble
A
pp
recia
ble
C
hang
e C
hang
e
-l--Y-
Wat
er
Tab
le
---
5D
~
F =
7
.85
; E
=
10
ksi
; D
= 1
7.7
5 ft.
; t
= 9
in
. u
nre
info
rced
; f'
m
c
4000
p
si;
Y
s
120
lbs/f
t3;
serv
ice l
oad
= 2
D
• Y
=
4.2
6 k
sf
s
Vert
ical
I M
ID
at
Loa
d at
I S
erv
ice
Cra
ckin
g
Lo
ad
k
sf
.31
3
2.3
94
.31
7
3.0
98
-
_._
--------~---. r-----.--.-. -
.32
5
4.2
6
J
Not
In
cre
ase
s A
pp
recia
ble
C
hang
e
un
it
wei
gh
t o
f so
il
1.0 ~
~-::
'.
80
70
60
..
Non
line
ar A
naly
sis
Exc
avat
ion
Loa
ding
¢=20
o
55
00
50
00
'" 5
0
. ...
....
....
·0
2SIZ
!0
_ " .. "
......
. 0 0
.. 0
····
3
41i!:
"CO
verp
ress
ure
Loa
ding
¢=20
o
20
00
50
.0
' 1
50
0
20
!0
2J
0
o 1
0
50
0
o I
10
£I
! £I
2£1
50
4
0
15
25
5
S
<5
Fl.
.CX
1BII
.!T
Y
AA
TIO
. F
FIGU
RE 3
.28
EFFE
CT O
F LO
ADIN
G CO
NDIT
ION
ON T
OTAL
ULT
IMAT
E LO
AD
. ":,r
-,>
" '.",
80
,-
__
__
__
__
__
__
__
__
__
__
__
__
__
__
__
__
__
__
__
__
__
__
__
__
__
__
--,
70
60
Lin
ear
Ana
lysi
s F
ull
Sli
p
\!m
=
0.4
5
35
00
30
e0
... u: S
0
" K
=0
.8
o /~===
25
00
_ " ..
~ o J 4
0
U ~ " ~ ;: 6
0
.J ., 2
0
10
K =
0.7
o
K =
0.5
o
Exc
avat
i on
Loa
dinq
20
00
!S0
0
, 0~~
50
0
o I
10
10
42
0 5
e
40
!
5 2
5
35
'5
rL
(XIB
ll.I
1Y
Q
1=I1
10.
f"
FIGU
RE 3
.29
EFFE
CT O
F CO
EFFI
CIEN
T OF
EA
RTH
PRES
SURE
ON
TOTA
L LO
AD
1/
'
"
this parameter could be near 1.0, while for firmer soils it may be on
the order of 0.5. The effect of the variation of K on the total load o
is shown in Figure 3.29; the strength in each case is obtained by using
a linear analysis and obtaining the ultimate load by projecting the
moment-thrust path to intersect the interaction diagram. At low
flexibility ratios the ultimate load is increased greatly by increasing
K but the difference 0'
cross at a flexibility
realistic value of K 0'
~rom zero to about 20.
decreases as K becomes 0
ratio of 30. Therefore
especially in the range
large
it is
of
until the curves
important to use a
flexibility ratios
Some of the gravity loading cases are quite severe as a result of
large moment at the crown while the corresponding thrust is small,
because horizontal forces result only from passive resistant due to
horizontal ovaling. The manner in which the gravity forces act on the
lining in any particular case, depends on the soil condition and the
excavation and support procedure. Several loading shapes were
investigated with the beam-spring model while varying the tangential
spr1ng stiffness relative to the radial values between zero and 40
percent. It is believed that the case with full slip or no shear
stress between the lining and medium is far too conservative for this
loading case and that a significant shear stress would always be
present. Four load shapes were considered: (1) a uniform vertical load
across the full lining, (2) only the radial components of a vertical
uniform load across the full lining acting on the lining, (3) uniform
radial load around the upper 180 deg of the lining and (4) radial load
over the upper 60 deg segment centered at the crown (See Figure 3.30).
The result of particular interest for design 1S shown in Figure
3.30 and contains all the data at failure of the lining in terms of the
ultimate thrust at failure (T ) divided by the axial failure thrust u
(To) plotted against the linear eccentricity (e~) divided by the lining
thickness (t). All the failure points fall in a rather narrow band and
provide a means of obtaining the strength that includes the nonlinear
behavior of the concrete by performing a linear analysis to obtain the
eccentricity e~ (by dividing the moment at the critical section by the
95
.~.,
.8
.7
.6
.5
~
a Unreinforced linings o Reinforced linings
Gravity Loading
Vertical uniform loading over full diameter
Radial component of vertical loading over 180 0
o .il H
E-< ~ p..:
E-! CJ)
;::J ~ ;:r:: t-<
.3
.2
. 1
0. 0.
.5 1 . 2.
.5 2.5
Radial uniform load over 60 0
o 0
3. 4. 3.5 4.5
o
Radial uniform load over 180 0
5. 5.5
o I
6.5 7.
FIGURE 3.30 PREDICTION OF FAILURE OF LININGS BY THE NONLINEAR ANALYSIS AS A FUNCTION OF THE LINEAR ECCENTRICITY DIVIDED BY THICKNESS
96
. " , ,
corresponding thrust). The criteria used to determine failure was a
concrete compressive strain of 0.004 or failure to converge in the
analysis if this strain was not reached. Failure by this criteria
always occurred after the interaction diagram was reached; for the
largest eccentricity, failure occurred on the interaction diagram but
for others the strain of 0.004 was reached shortly after the
moment-thrust path turned inward after leaving the interaction diagram;
normally there was some capacity of the lining remaining in the latter
cases that was larger as the flexibility ratio increased. Also failure
occurred at a higher thrust after leaving the interaction diagrams for
the unreinforced lining than for the reinforced linings; that is, there
was more reserve capacity remaining. The variation of compressive
strain for a typical reinforced lining is shown in Figure 3.31 where
the interaction diagram and moment-thrust paths are shown with the
compressive strains indicated on the paths.
Figures 3.32 and 3.33 compare the var10US loading conditions in
terms of p/f' and flexibility ratio (F). The pressure p for the c
excavation loading and overpressure loading are the full overburden
pressure at the level of the tunnel rather than that acting on the
lining (part of this pressure arches around the lining), while that for
the gravity loadings is the pressure of a certain height of soil above
the lining acting directly on the lining. Though these loading are not
directly comparable it is convenient to place them on the same plot and
instructive to compare them. It is clear that if full overburden
pressure 1S used for the design for all the loading conditions, the
gravity loadings would always govern, but this would be reasonable only
for very shallow tunnels. For deeper tunnels the gravity loading would
be limited to some depth of soil above the tunnel on the order of two
diameters 1n the worst case and less in most cases as discussed in
Chapter 1.
The excavation and overpressure loading cases were analyzed with
joint elements between the linear medium and the lining to represent
the shear stress on this surface. In the gravity loading cases where
tangential springs are used to represent this shear stress, four
97
U:l p.. H ~
~
E-< U:l
~ ::r:: E-<
200.
180.
160.
140.
120.
100.
80.
60.
40.
20.
0.
Radial component of vertical load applied over upper 180 deg Reinforced with 0.5 percent each face Kt/Kr = 0.25
f' c
f r
R t
4000 psi (28 MPa)
500 psi (3.4 MPa)
118 in. (3.0 m) 10 in. (250 mm)
0. 100. 200. 300. 400. 500. 600. 700. 50. 150. 250. 350. 450. 550. 650.
MOMENT, IN-KIPS
FIGURE 3.31 COt4PRESSION STRAIN VARIATION ALONG THE MO~1ENT - THRUST PATH
98
·-.1
2
• 1
~
C)
4-l
"- p. ~
.RJ8
0 H
~ ~
\0
~
\0
P:::
.121
8 ::;
:J U
) U
)
~
P:::
p....
.121
4
.121
2
121.
.0
EX =
Ex
cav
ati
on
lo
ad
ing
O
P =
Ov
erp
ress
ure
lo
ad
ing
RC
180
=
Rad
ial
Com
pone
nt
of
vert
ical
load
ing
o
ver
18
00
R60
=
R
ad
ial
un
ifo
rm
load
o
ver
60
0
R18
0 =
Rad
ial
un
ifo
rm
load
ing
o
ver
18
00
VT
=
Vert
ical
load
acro
ss
full
d
iam
ete
r
--
Rein
forc
ed
--
-U
nre
info
rced
K
/Kt
= 0
1121
21
21
FL
EX
IBIL
ITY
,F
FIGU
RE
3.32
,
.J.2
K /
K
0.1
25
t
r
.1
~
C)
4-l
"- P., ft
.121
8 0 H
~ ~ ~
P:::
.121
8 ::;
:J U
) U
)
~
~
p....
.. 121
4
.121
2
121.
4121
2J
21
21
4121
51
21
1121
F
LE
XIB
ILIT
Y,F
'5
.0
COM
PARI
SON
OF
LOAD
ING
COND
ITIO
NS
FOR
Kt/K
OF
0 A
ND 0
.125
FO
R GR
AVIT
Y LO
ADIN
G IN
TER
~lS
OF p
/f'
r c
. 1
2
• 1
-u
lH ---P.. ~
0 .0
8
H
E-< ~ '"-l
I-'
p::
0 ;::
J 0
(/l
.06
(/l
'"-l
p::
P-,
.04
.02
0.
o
.... ,),
. ,I
'.
EX
=
Ex
cav
ati
on
lo
ad
ing
OP
=
Ov
erp
ress
ure
lo
ad
ing
R
180
=
Rad
ial
un
ifo
rm
load
ing
o
ver
18
00
--
Rein
forc
ed
---
Un
rein
forc
ed
K
/K
=0
.25
t
r
----
R60
20
1
0
FL
EX
IBIL
ITY
F
30
,
40
-U
lH
---P 0 H
E-< ~ '"-l
p::
;::J
(/l
(/l
'"-l
p::
P-,
.12
• 1
.08
.06
.04
.02
0.
o
RC
180
=
Rad
ial
Com
pone
nt
of
vert
ical
load
ing
o
ver
18
00
R60
=
Rad
ial
un
ifo
rm
load
o
ver
60
0
VT
=
Vert
ical
load
acro
ss
full
d
iam
ete
r
K /
K
0.4
0
t r
20
4
0
10
F
LE
XIB
ILlT
Y,F
3
0
FIGU
RE
3.33
CO
MPA
RISO
N OF
LO
ADIN
G CO
NDIT
IONS
FO
R K
t/Kr
OF 0
.25
AND
0.40
FO
R GR
AVIT
Y LO
ADIN
G IN
TER
MS
OF
p/f
' c
--
different values were investigated and it is clear from Figures 3.32
and 3.33 that increasing the tangential spring stiffness relative to
the radial stiffness increased the lining strength, and the 1ncrease 1S
greater as the flexibility ratio increases because the absolute value
of the spring stiffness is also increasing.
Figures 3.32 and 3.33 also show that reinforcement increases the
pressure on a lining required to cause failure, because the
reinforcement increases the thrust capacity, but this thrust capacity
can easily be replaced by a very small 1ncrease in concrete strength or
thickness. The curves of Figures 3.34 and 3.35 show the data in the
previous figures except that the flexibility ratio 1S plotted against
T /T defined earlier, and the difference between the reinforced and u 0
unreinforced cases is reduced. This shows that the reduction in thrust
capacity by moment 1S essentially the same for a reinforced and
unreinforced lining.
The results for the vertical, radial component of vertical and
radial over 60 deg loadings are shown 1n Figure 3.36 in terms of
p/f' vs F. c
Above F
Here it is clear what effect the tangential shear has.
20, for example, the shear spring stiffness of 40 percent
of the radial values can increase the lining capacity by 50 to 100
percent. The effect of reinforcement is also shown, and its influence
appears to be constant throughout the range of F except for val~es
below about 10, where the capacity of the unreinforced lining appears
to decrease more rapidly for the radial component of vertical.
Cracking in linings can be evaluated by studying the strains that
occur at critical sections during loading; in Figures 3.37 and 3.38
contours of tensile strain at the tension face of the concrete are
shown on the moment-thrust paths for the radial component of vertical
loading, K /K = 0.25 and the reinforced and unreinforced linings. t r
From these curves the thrust level can be seen relative to the ultimate
thrust for a particular strain reached. Flexural cracking is generally
considered to occur at about 0.00015 strain for a fairly rapid loading,
but if the loading is slow it is reasonable to double this. Therefore,
Wl
"
.", . . ,~
.8
.7
E-!°
.s
........ E-!=
='
a
0 .5
H
E-
! ~ E-!
Cf.l
.4
r-'
p
0 ~
N
::r::
E-!
.'3
.2
• J
~.
o
:'.: -!
,.
R60
=
Rad
ial
un
ifo
rm
load
o
ver
600
R18
0 =
Rad
ial
un
ifo
rm l
oad
ing
o
ver
180
0
RC
180
= R
ad
ial
com
po
nen
t o
f v
ert
ical
load
ing
o
ver
18
00
.8
VT
=
Vert
ical
load
ing
acro
ss fu
ll
dia
mete
r
--
Rein
forc
ed
--
-U
nre
info
rced
K
/K
=
0
t r
1--0
.~--0
~-
• .,.-
p ,,- '3R
J
.,.,. ..
...... .
VT
40
K /
K
0.1
25
t
r
.7
E-!
°.s
.....
... f-l
=='
A 8 .5
E-
! ~ E-! g3
.4
~
::r::
E-!
.3
.2
• 1
0.
o 2
0
10
F
LE
XIB
ILIT
Y,
F F
LE
XIB
ILIT
Y,
F
FIGU
RE 3
.34
COM
PARI
SON
OF G
RAVI
TY
LOAD
ING
SHAP
ES
FOR
Kt/
K
OF 0
AND
0.1
25
r
4-0
'30
'," ..
.8
.7
.S
0 E-
I --;:l E-I
.5
~
0 H
E-<
~
<
0 p:
; .4
V
ol
E-I
UJ ~ ::r:
"E-<
.?i
.2
• 1
0.
o
R60
=
Rad
ial
un
ifo
rm
load
o
ver
60
0 R
180
= R
ad
ial
un
ifo
rm l
oad
ing
o
ver
18
00
RC
180
= R
ad
ial
com
pone
nt
of
vert
ical
load
ing
o
ver
18
00
VT
= V
ert
ical
load
ing
acro
ss fu
ll
dia
mete
r
--
Rei
nfo
rced
--
-U
nre
info
rced
K
/K
=
0·2
5
t r
2.0
4
.0
.8
K /
K
0.4
0
t r
.7
.6
0 E-
I --;:l E-< .5
M
0 H
E-< ~
4 E-
< •
UJ ~ ::r:
E-<
.5
.2
• 1
0.
.0
2.0
1
0
FL
EX
IBIL
ITY
, F
""3.0
J.
0 F
LE
XIB
ILIT
Y,
F 31
1')
FIGU
RE
3.35
CO
MPA
RISO
N OF
GRA
VITY
LO
ADIN
G SH
APES
FO
R K
t/K
OF 0
.25
AND
0.40
r
4iJ
, ("'6
,045
.03
.015
0.
.06 ~ U 4-i
---- .045 p.
~
0 H .03 H
~ ~ .015 p::: ~ C/J C/J 0. ~ p::: ~
.06
.045
.03
.015
0.
0
0
0
Vertical uniform loading over full diameter
10 20 30 5 15 25
FLEXIBILITY RATIO, ~
Radial component of vertical loading over 180 deg
10 20 5 15 25
30
FLEXIBILITY RATIO, F
Radial uniform load over 60 deg
40 35
40 35
10 20 30 40 5 15 25 35
FLEXIBILITY RATIO, F
Reinforced D K /K 0 /', K /K t r t r Unreinforced 0 0.125 <>
45
45
45
0.25
0.40
FIGURE 3.36 EFFECT OF Kt/Kr
FOR THE VARIOUS GRAVITY LOADING SHAPES
104
,,,- ',j
.,
::-:
C/l p.,
H ~
~
H
C/l
I-'
~ 0 V
1 f5
20
0.
17
5. I
15
0.
12
5.
10
0.
75
.
50
.
25
.
0.
Rein
forc
ed
wit
h
0.5%
in
ea
ch
face
t =
10
in
. (2
50
1 mm
) R
'~~118in. ~3.0
m)
f'
= 4
00
0 p
si
(28
MPa
) c
e: =
t
F=
l. 7
4
0.
20
0.
40
0.
00
0.
10
0.
30
0.
50
0.
70
0.
MOM
ENT.
IN
.-K
IPS
FIGU
RE
3.37
TE
NSIO
N ST
RAIN
S AL
ONG
THE
M-T
PA
TH
FOR
A R
EINF
ORCE
D LI
NING
"
.,
C/l p.,
H ~ ..
H
C/l
::;J r:r:
::r::
H
2fM
3.
Un
rein
forc
ed
F 3
0 'c
o
17
5.
15
0.
E
12
5.
t
0.0
00
6
10
0.
75
.
50
.
25
.
0.
0.
20
0.
40
0.
00
0
10
0.
30
0.
50
0.
'70
0.
MOM
ENT.
IN
. -K
IPS
FIGU
RE
3.38
TE
NSIO
N ST
RAIN
S AL
ONG
THE
M-T
PA
TH
FOR
AN
UNRE
INFO
RCED
LI
NING
the 0.0003 contours in these figures may be considered to approximately
indicate cracking in linings. In that case cracking occurs at about 25
to 30 percent of the ultimate thrust for the reinforced lining for the
range of F shown; for the unreinforced lining the cracking thrust ~s
about the same level relative to the ultimate for the two larger values
of F, and is about 40 percent for the smallest F. The range rema~ns
the same (25 to 30 percent) for the reinforced linings for all loading
conditions, and for the unreinforced lining it is between 29 and 37
percent for the larger values of F and 40 to 50 percent for the
smallest.
After cracking the width of cracks can be estimated. For the
reinforced lining a strain of 0.0006 at the critical section and crack
spacing of 10 in. (254 rom) would correspond approximately to a crack
width of 0.006 in. (0.15 rom), and the thrust level at which this would
occur can be observed on Figure 3.37. The width would likely be larger
for the unreinforced lining because the spacing would be larger.
Effect of Reinforcement: Three different reinforcement ratios (0,
0.5 and 1.0 percent) were used to investigate their effect on the
lining behavior. The moment-thrust paths of the critical sections for
failure are shown ~n Figure 3.39 for different flexibility ratios. It
will be noted that the paths for different reinforcement ratios do not
change ~n the linear range. Although they approach their respective
envelopes showing higher ultimate thrusts for higher reinforcement
ratios, the total ultimate load does not vary appreciably (Figure
3.40). The moment-thrust paths for the critical section for cracking,
which ~s the crown, is plotted ~n Figure 3.41, which also includes the
cracking envelopes according to cracking strains of 0.00015 and
0.00030. Since the cracking envelope remains essentially the same for
all three reinforcement ratios, the cracking loads are also not
affected by variation in reinforcement. This diagram also points out
that cracking may be a problem for low flexibility ratios. The same
trend has been noticed for excavation loading also. However, if the
cracking strain is increased to 0.0003 as a result of creep and the
slow application of load, cracking is greatly reduced. Only below a
106
(f) a..
MOMENT (KN-M)
o 20 40 60 80 600. r--------+--------r--------r--------~---
550. Unreinforced EnveloDe
Overrressure Loadina <p =: 20v; c =: 3.5 psi 2500 (joint element prop.) t = 10 in. (2S4mm)
500.
450. Reinforced Envelopes 1.0';
400.
~ 350. 1500
F
<f 250. ...... x a:
1000
150.
500
50.
0. ~~~--_+~~~-L+_--~---+---~--~0 0. 200. 400. 600. 800.
MOMENT (IN-KIPS)
FIGURE 3.39 EFFECT OF REINFORCEMENT ON MOMENT-THRUST PATHS FOR OVERPRESSURE LOADING
107
z ~
I-(J)
::J ell: I I-
...J a: H
X a:
u.. (j)
":d.
o {[
70
60
50
040 .J
W U {[ u.. iY :J 30 (j)
L iY o u.. H
Z :J 20
10
o
Overpressure Loading
~ = 20°; c = 3.5 psi; (joint element prop.)
v = 0.4 m
At Cracking Strain = 0.0003
Cracking Strain = 0.00015
10 20 5 15 25 35
FLEXIBILITY RATIO, F
+ 33012
T 5000
1
r 2700
12400 ,
IT 0... , /
- 2100
I 0 c 0 ; ...J
lJ -.-.. ~ .-
f
~) J)
'"" c'-l. 1200 n
Lc. > ...
--
60QJ
3QJQJ
40 45
FIGURE 3.40 EFFECT OF REINFORCEMENT ON TOTAL LOAD AND CRACKING LOAD
108
. ~
, --"." '
C~OWN MOMENT (KN-MJ 39 50 70 90
2£J 40 Gil 89 600. ~--4---~---+---4----~--+---4---~--~
(f)
0..
550.
500.
450.
400.
:: 350.
<E 250.
1 ....... x a:
200.
150.
1£10.
50.
0. 0.
E~ve lopes for Cracking Strilin
0.00015
F =
200. 1130. 3fm.
Overpressure Loadinq
¢ = 200; c = 3.5 psi;
v = 0.4 rn
Envelopes for
400. 6.0.0. 5.00. 7.00.
CROWN MOMENT (IN-KIPS)
2500
1500
FIGURE 3.41 EFFECT OF REINFORCEMENT ON FIRST CRACKING FOR OVERPRESSURE LOADING
109
flexibility ratio of about 10 is flexural cracking likely to be a
problem, and the relative values of cracking load and ultimate load are
shown in Figure 3.40 in terms of uniform surface pressure. The effect
of doubling the cracking criterion ~s shown, where the ultimate load
shown is the actual ultimate based on the nonlinear analysis. Minor
cracking due to flexure is not likely to create leakage because the
compression zone of concrete remains an effective barrier to water
passage. However, since reinforcement does help in reducing crack
widths by distributing them, it may be advisable to use it ~n the
amount required from the cracking point of view if it is calculated to
occur and will create problems. An addition of a reasonable amount of
reinforcement does not appreciably improve the load carrying capacity
of the lining, however.
Effect ~ Joints in Segmented Linings: The presence of joints in
segmented linings causes reduction ~n the stiffness of the lining,
which in turn increases the flexibility ratio and thus reduces the
moments in the li@G This reduction in the lining stiffness was obtained
from the analysis for a set of typical dimensions by calculating an
equivalent modulus of a monolithic lining (E ) that will give the same 2 eq
moment coefficient (M/pR ) as given by a segmented lining of the same
thickness. Using the closed form analytic solution described by
Ranken, Ghaboussi and Hendron (1978) for linear analysis with
excavation loading and full slip conditions, the moment coefficients
for different values of elastic moduli of the lining (E 9,) and of medium
(E ) were obtained for a monolithic lining of 8 ~n. (200 rom) thickness m
and 19 ft 8 ~n. (6 m) diameter (Figure 3.42). Two types of segmented
linings with eight and four segments per ring with a joint always at
the crown were analyzed using the beam-continuum model. The joints
were represented by very short unreinforced beam elements with concrete
stress-strain properties without tension. The maximum moment
coefficient obtained for the lining from the analysis of a segmented
lining ~s entered in Figure 3.42 to obtain an equivalent modulus of
elasticity of the lining (E ) that would give the same moment
coefficient for a monolithic eg
lin~ng. The ratio of this modulus to the
original modulus of elasticity of the segmented lining is plotted in
llO
MODULUS Dr THE LINING. Et (MPa) 5,000 10,000 15,000 20,000 25,000
.~7r------+------~------~-----+------~--~ Excavation Loadinq; ~ull Slip
I
.~S I
..; .05 (5 .... IU W tJJ
...J (.C U
;: .• 94 ..... 0/: U
Ia: Iz W .... ~ _.35 LL.. LL.. W o U
IZ W § ~ .32
.al
K 0.5 o R 118in.(3 m)
t = 8 in. (200 mm)
E = 1000 ps i m
/ /
(6.9 MPa) •
2500 psi (17.2 MPa)
5000 (34.5 MPa
-
7500 psi (51.7 MPa)
v S. ,;~---+----4-----~---+----4---~~---+----4
5E0~ 1500, 2SB0. 3500. 110DULUS or THE LINING., Et (ks i )
FIGURE 3.42 MOMENT COEFFICIENT VS MODULUS OF LINING FOR AN 8 IN. (200 mm) MONOLITHIC LINING IN DIFFERENT SOIL MEDIA
111
. I
MO
DU
L.U
S O
F
TH
E
ME
DIU
M,
Em
XS
""
"
(KP
A)
10
!S
"
6"
7"
q
0
I1J
:20
4
0
6"
a"
1 ""
s ...
I I
I
Thi
ckne
ss =
8 i
n.
....
-1-~~
'"-t
= 1
2 in
. F
= 4
.0
(30
0
mm
) .8
4-
/ /'
----
----
~ /
.7 "
'" /
4"
/'
Fle
xib
ilit
y r
atio
s w
here
jo
ints
beco
me
inef
fect
ive
in
redu
cing
.8 +
/
/ ~
-..
= 1
0 in
. EQ
,
(25
0
mm
) c<
u
.J
.. 6
.....
... 0-
f-'
OJ
f-'
w
.4 +
/
/ /
R =
11 8
i n . (
3 m
) N
Exc
avat
ion
Loa
ding
~~ t
/ //
Ful
l S
lip
K
0.5
0
.:2
"'"
r//
8 se
gmen
ts
per
ring
, • I , , I #
, #
.1"
,"
/.,'
#
, ,
I #
I 1
1/
,-,:<1
..
+ 0
. ..
" :2
4
8 a
10
1:
2 1
4
1 ~
5 7
<\
11
1
5
16
M
OD
UL
US
or
TH
E
ME
DIU
M,
£m
X1
0"
"
(PS
I)
FIGU
RE
3.43
EQ
UIVA
LENT
M
ODUL
US
OF
ELA
STIC
ITY
OF
SE
GMEN
TED
LIN
ING
S
\"':'::
, '
.. {
'."
Figure 3.43 for different medium moduli and lining thicknesses. Almost
identical curves were obtained for both four and eight segments per
ring with slightly higher values of E /E obtained in the lower range eq
of medium modulus for four segments per ring. These results show that
the stiffness of the segmented linings in the practical range of soft
ground could be reduced by the effect of the joints to 30 to 95 percent
of that of a monolithic lining with the same thickness. Once the
modulus of the medium reaches a certain value, however, the joints
become ineffective in reducing the stiffness and thus the moments 1n
the lining (Figure 3.43), because of high thrust values that do not
allow the joints to open. Thus, for certain combinations of lining
thickness and medium modulus, segmented linings could be treated as
monolithic from an analysis point of view. The number of joints per
ring did not significantly influence the magnitude of moments in the
lining for the particular joint orientations selected. It is, however,
noted that the values of E /E eq
used as a guide in actual design
assumptions made 1n obtaining
shown in Figure 3.43 should only be
problems, because of the numerous
them (i.e., modeling of the joints,
excavation loading, K value specific radius of the opening, and o '
specific joint orientation). Nevertheless, they provide an indication
of the effect of joints, in conjunction with other parameters, on the
lining stiffness and the order of magnitude of the reduction to be
expected.
3.2.3 Parameter Study for Circular Linings 1n Rock
The beam-spring model used for the study of arches in rock and
described in Section 3.2.1 was modified and used to examine the effects
of various parameters when a loosening rock load is applied directly to
a circular lining. The model used 1S shown in Figure 3.44 and a
uniform loosening load was applied across the lining.
Parameters studied were the relative flexural stiffness of the
medium to the lining described by the flexibility ratio F, relative
stiffness of the tangential and radial springs K /K radius to t r'
thickness ratio of the lining R/t, and the lining reinforcement. A
113
) l .•
Force Nomenclature for Each Element
R = 1 06. 5" ( 2 . 7 m) No.
No.
No.
No.
of Elements = 25
of Joints = 51
of Equations = 99
of Radi a 1 Spri ngs = 26
No. of Tang. Springs = 24
12"
I~ (3~ 0.5% Steel at 9" (229 mm) Lk.·· .. ·.··~ each face
f~ = 4000 psi (28 MPa)
f = 40,000 psi y (276 MPa)
FIGURE 3.44 FINITE ELEMENT MODEL OF CIRCULAR LINING FOR PARAMETRIC STUDIES IN ROCK FOR LOOSENING LOAD
114
particular typical lining that was 9 in. (230 mm) thick with a radius
of 106.5 in. (2.70 m) was used for most of the study, and then the
thickness and reinforcement of the lining were changed to study the
effects of Rlt and reinforcement. The reinforced concrete section
contained concrete with compressive strength of 4000 psi (28 MPa) and
0.5 percent deformed bar reinforcement 1n each face with a yield
strength of 40 ksi (276 MPa). The concrete strength was changed to
4310 psi (30 MPa) for the unreinforced section so the axial thrust
capacity would be the same as for the reinforced one 1n order to
examine the effect of reinforcement on other aspects of behavior such
as moment redistribution and ductility. The beam element used to
represent the lining can have nonlinear behavior that depends on the
material properties. The Hognestad equation was used for the
compressive stress-strain curve for the concrete and the reinforcement
stress-strain curve was elastoplastic.
Relative stiffness of the medium to the lining 1n flexure was
determined by
Mohraz (1972),
the flexibility ratio as presented by Peck, Hendron and
and the relationship between the radial spring
stiffnesses and the elastic modulus of the medium given by Dixon (1971)
was used. The formulas for these relationships are given in Section
4.3.4 of this report. The tangential spring stiffness was a variable.
The most important parameters affecting lining strength are the
medium stiffness and tangential to radial spring stiffness ratio; the
effects of these parameters on the lining strength 1n terms of the
thrust ratio T IT is shown 1n Figure 3.45. If the lining has a 10 in. u 0
(250 mm) thickness and 19 ft (5.8 m) diameter, and is 1n a very soft
rock with a Young's modulus of 100,000 PS1 (690 MPa) the F would be on
the order of 70, provided the lining 1S uncracked. This 1S the range
shown by the curves that hoe a steep slope so the T IT ratio can u 0
become low and 1S sensitive to the medium modulus selected. In this
range care must be exercised in selecting the loading and the medium
properties. If the lining cracks, its stiffness is immediately reduced
relative to that of the medium, and if the cracked moment of inertia is
one-half the uncracked value for example, the value of F is doubled and
115
1 .
.q
.8
.7
.6
0
I-'
l-I-
' 3
.5
0\
r-.4
.3
.2
.. 1
0.
.~:
0.
';, ~
R/t
=
14
-----------t-~
K t /
K
=
25%
_
__
_ ---/
~-.-
--~-
-r
0 -r
.. o.
----
-_~-
----
----
----
-~-:
:t~1
_
__
__
_ ------12~X-~-<
R/t
=
1
1.8
8% -
----
-y.
0%
0
Rei
nfor
ced
0.5%
ea
ch
face
1.
2.
3.
4.
5.
6.
7.
.5
1.5
2
.5
3.5
4
.5
5.5
6
.5
F X
1
00
0
FIGU
RE
3.45
LI
NING
CAP
ACIT
Y AS
A F
UNCT
ION
OF
FLEX
IBIL
ITY
RA
TIO
shifts to the right on the curves increasing the strength of the lining
appreciably. When F 1S larger than 300, which corresponds to a
deformation modulus of the medium of about 500,000 psi (3450 MPa) for
the particular lining considered, the curves become rather flat,
indicating a greatly reduced sensitivity to the medium modulus. In all
cases analyzed in Figure 3.45, a plastic hinge occurred first at the
crown and failure of the lining was precipitated by conditions at this
location, though hinges started to develop at other location as well
prior to collapse. In Figure 3.46 the vertical axis of Figure 3.45 is
changed to show the actual uniform pressure on the lining at failure,
where the definition of failure has been selected at a strain of 0.004
in the concrete. This is a conservative estimate of failure, as more
load could be resisted beyond this point in most cases. The shape of
the curves is very similar to that in Figure 3.45.
The effect of tangential spring stiffness is more easily shown on
Figure 3.47 where the T IT ratio 1S plotted against the spring u 0
stiffness ratio for various values of F. These curves show that there
1S very little change in strength for all values of F when K IK is t r
larger than about 0.12. Most designers agree that the effective
K IK in the actual tunnel in rock t r
analysis is not greatly sensitive to the
is at least this large, so the
value selected. The range
normally used is from 0.10 to 0.50. If the value selected is smaller
than 0.12 because the opening walls are smooth or a large amount of
wood blocking remains between the final lining and the rock, then the
effect of the value selected depends on the flexibility ratio; for low
values of F the sensitivity remains small, and becomes larger as F
increases.
The lining-medium system 1S not defined completely by the
flexibility ratio, as shown 1n Figure 3.45 by the three curves with
different radius to thickness ratios R/t with K IK of 0.25. The t r
separation of the curves shows the effect of the R/t ratio. The reason
for the variation with R/t is the effect of thickness on the nonlinear
behavior of the lining, where the thicker lining has a larger
T IT ratio at the same F. This variation of the strength of the u 0
117
50
~o
l.!-
UI
:L
Cl
« G
-.I
30
-.I
« U
~--
I-'
cr::
I-'
w
00
>
-L a:
0
20
LL
>
-<
z:
::J
10
:~ -'
.
, "
~--------------------------------------------------------------------T+ 2
.5
(/
( ,I'
cl.
K/K'
=25~
:'
,-8
t
r
-'-
.---.-"
12.5
/~
0°,' /U
r,/
I __
_ e
--
__ -
---
------
---
---
----
---
---------
----
----
----
---
----
_€
l
/,/ 1/': " I .1
, , .. .. ,
1.0
R/t
11
.8
Rei
nfo
l'cei
llen
t G
.5:;
2.0
3
.0
4.0
FLEX
IBIL
ITY
RA
TIO,
F
X 1
000
FIGU
RE
3.46
TO
TAL
LOAD
VS
FLEX
IBIL
ITY
RA
TIO
'i
"
5.0
6
.0
2.0
1.5
1.0
0.5
n:l
0-
Cl
c(
o -.I
-.I
« u I a:
w
:>
c::
o l.!- :::.:
::J
1 .
.9
.8
.7
0 t:.6
::
l ~
r-a
~.5
r-a
\C
~
Cl:
0::: ~.4
Vl
:::>
0::: ~.3
. 2
. 1
0.
:., ..
..
• U
nre
info
rced
--
Rei
nfo
rced
(0
.5%
)
----
----
--+
----
F =
62
30
F =
12
50
,r"F
=
75
/ ~.
F =
7
.5
• 1
0 L
0
0.
.04
.0
8
.12
.
16
.2
.2
4
SPRI
NG
STIF
FNES
S RA
TIO,
K
t/Kr
FIGU
RE 3
.47
LINI
NG
CAPA
CITY
AS
A F
UNCT
ION
OF T
ANGE
NTIA
L TO
RA
DIAL
STI
FFN
ESS
RATI
O (K
t/Kr)
FO
R R
/t =
11.
8
lining as depicted by the thrust ratio 1S negligible for all practical
purposes when the flexibility ratio is higher than 1250. However, for
lower values of flexibility ratio, the radius to thickness ratio (R/t)
can make appreciable difference as shown in Figure 3.45. If the radius
and flexibility ratio are kept constant in a linear analysis, variation
of the R/t ratio by changing the thickness of the lining does not
affect the thrust ratio when the moment-thrust paths reach the envelope
above the balance point; however, near the balance point the thrust
ratio can become smaller for thicker linings while below the balance
point it increases with the lining thickness. If a nonlinear analysis
1S performed, the effective flexibility ratio due to cracking and
nonlinear stress-strain properties of the lining, will be increased
more drastically for the thicker than for a thinner lining, so the
thrust ratio is increased more for the thicker lining. Thus a thicker
lining for the same radius and initial flexibility ratio, but lower R/t
ratio, gives a higher thrust ratio.
The moment-thrust paths combined with the failure envelope for the
critical sections in Figure 3.48 show how the flexibility ratio affects
strength. When the flexibility ratio 1S small and therefore the
deformation and moments are large, the moment-thrust paths reach the
failure envelope below the balance point. The concrete and steel 1n
the section are fairly ductile so the moment-thrust path follows the
failure envelope until the concrete starts to crush on the compression
side of the section. When crushing occurs, the internal thrust
resultant shifts inward toward the center of the section, reducing the
lever arm and therefore the moment, but the thrust can continue to
1ncrease as more concrete toward the tension side starts to resist
compression. The reduction in moment causes the moment-thrust path to
turn inward toward the thrust aX1S as shown for F = 7.5 and 75. The
maximum thrust and thus the peak load is obtained when the rotational
capacity of the critical section is finally reached. A value of F = 70
1S about the lower limit for linings in rock and for this value the
path normally approaches the interaction diagram near the balance
point.
120
MOMENT (KtJ-M)
--""'(:'''l ~ c:: L: iJ
~ r. -,. ,r-{
,/ 'Lil' 'I' _ L C-'
Rjt = 11.833
,100. F =
~~1oment-Thrust Envelope 1 Ql7r~ '~,--' ~J
350. 1 ('. /":,~ C> 'L..l"/...)
'I 1 -r.-"
~ !.. I:...' '0
~ ,300. c.. ~ F
Z
1 2Gi -' ~ {~ lil '-"
I- 250, V)
::> 0:::
l-V)
::> 0:::
::r:: I- l'00:J :::c
I-
...J ...J « 20O, >< «
« ><
8 i2)21 «
150. 6°1' '~I 'i)
100, L11.2 [I
F = 7.5 --, ~-. '/L,II' _ j~J
0. 0. 100. 200. 300. S00.
MOMENT UN.-KIPS)
FIGURE 3.48 EFFECT OF F ON MOMENT-THRUST PATHS AT CROWN
121
'. :..-: . .;~. ',,.\
When the flexibility ratio LS large, as for F = 1250 and 6230 Ln
Figure 3.48, the lining deformation and therefore the moment at the
critical sections are smaller, so the moment-thrust path reaches the
failure envelope above the balance point where the rotational capacity
of the section LS much smaller due to the large thrust. In this case
the limiting rotation of the section LS reached when the failure
envelope is reached and there LS no following of the envelope, but the
thrust and therefore the load on the lining is considerably larger.
Since the ultimate load is nearly proportional to the thrust
capacity, and the thrust capacity depends on where the moment-thrust
path reaches the failure envelope, Figure 3.48 also shows how the
flexibility ratio affects strength of the lining. The moment-thrust
paths start as if the problem solution is linear, before the nonlinear
effects begin. If the initial path LS projected linearly to the
intersection with the failure envelope the linear prediction of failure
thrust would be obtained. Therefore, this figure can be used to
visualize the difference between the linear and nonlinear prediction of
failure, and it can be seen that above the balance point a linear
prediction of the failure thrust is much more accurate than below the
balance point. It also shows that the relationship between the
stiffness of the medium and intersection of the moment-thrust path with
the envelope is highly nonlinear.
The moment distribution around the lining at the maXLmum load LS
shown in Figure 3.49 for several values of F. Peak moment of opposite
sign occur at the crown and at about 45 deg from the crown. For F
7.5 and 75 the moment path reaches the failure envelope below the
balance point (Figure 3.48) where an increase in thrust Lncreases the
moment capacity so the moment is larger for the larger F; however, for
the two larger values of F an increase in thrust reduced the moment
capacity above the balance point, so the moment is smaller for the
larger F. Although peak moment at about 45 deg from the crown may
become slightly larger than that at the crown, the thrust at this
section is higher, so the crown still governs the design. In the lower
122
portion of the lining the moments are fairly small and are not likely
to govern the design.
Internal shear distribution induced in the lining is shown in
Figure 3.50 for the maXimum loads. There are regions of peak shear
near 20 deg and from 45 to 60 deg from the crown. The shear is larger
at 45 to 60 deg for the high flexibility ratio and at 20 deg for the
low values. The section closer to the crown are likely to be more
critical, however, because the direction of diagonal tension at this
section leads to movement into the tunnel of the loaded region near the
crown.
Most of the solutions obtained for these comparisons were obtained
for a lining with 0.5 percent reinforcement in each face, and then
several problems were worked for a lining without reinforcement, but
with f increased to compensate for the loss of axial thrust by removal c
of the reinforcement. The T /T ratios are compared for the reinforced u 0
and unreinforced linings in Figure 3.47 where there 1S essentially no
difference. Reinforcement had a negligible effect on the stiffness of
the lining for the larger values of F and increased the stiffness only
slightly near failure for the low values of F. Moment-thrust paths are
compared with and without reinforcement and with the failure envelopes
in Figure 3.51. The envelopes are the same in the high thrust range
but different below about 300 kips (1330 kN) where the moment capacity
is increased by the reinforcement. There is little difference in the
moment-thrust paths except for the lowest F; in this case the paths
start to separate at a thrust of about 90 kips (400 kN) and each path
approaches its respective envelope; the maXimum moment that the
reinforced section can resist is larger, but both paths turn back
toward the thrust aXiS and reach essentially the same peak thrust so
the load on the lining is essentially the same in the two cases.
Therefore, in the range of flexibility ratio appropriate for linings in
rock, reinforcement in the lining adds little strength or ductility in
circular linings. Reinforcement is needed only in some cases for
serviceability considerations, provided the design loads actually
124
if' '
occur, because at serV1ce loads the thrust 1S large enough to prevent
appreciable tensile stresses.
125
:z: ::,.L
OL G LJ -(F,
60
40
20
o
-20
-40
-60
-60
-40
-20
o
20
40
60
(J)
D.....
::,.L
CL G W ....L (F,
15 ~
15 ·4
I -7.5 .
O-~ 15f
I 15 ..
Kt/K r = 12.5%
R/ t = 11.3
FIGURE 3.50 EFFECT OF F ON SHEAR AT MAXIMUM LOAD
126
50
0
L1
50
L1
00
55
0
(/)
~ 6
01
0
:.c
I-'
N
I-":
l C
C?1
~
c:..
;::)
L
-...J
a::
:
--:r
: I-
' l-
N
....
12
00
ex:
> <
{
x <{
15
0
10
0
S0
(2)
MO
MEN
T (K
N-M
)
10
3
0
50
0
20
4
0
60
--------I---,-,--------+---------4---------"~------~----_1
22
00
( K t /
K r
= 2
5%,
F =
62
30
)
_ (K
t/K
=
12
.5%
, F
= 6
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CHAPTER 4
DESIGN RECOMMENDATIONS
4.1 SCOPE
It is well known that concrete and masonry linings in tunnels have
the capability of supporting substantial loads, even though
unreinforced, and even though the ground loads and distortions may be
sufficient to cause tension cracks to form and open. As a tunnel
lining in continuous contact with the soil deforms and approaches its
bending capacity, it will continue to build passive reaction against
the soil or rock that will prevent the unrestricted deformation,
increasing eccentricity, and collapse that would occur 1n a simply
supported column. As bending cracks occur and the passive reaction
continues to increase, the eccentricity of thrust in the section
actually decreases and the lining section continues to be able to take
higher thrust. The resulting thrust capacity of the section is
significantly greater than would be obtained for a simply supported
column, particularly in the case of unreinforced sections. The level
of the ultimate thrust that is finally reached 1S influenced by the
amount of damage that takes place in the section as the lining deforms.
Although it has been recognized that a concrete tunnel lining can
reach limiting moments without collapse, there has been little
information on the maximum load levels that a lining can sustain under
such conditions. In the research studies that have led to this report,
large scale model studies and nonlinear analyses were carried out on
concrete tunnel linings, both reinforced and nonreinforced, in ground
having stiffnesses ranging from soft soil to hard rock. The studies
concentrated on evaluation of the ultimate post cracking, nonlinear
behavior of the concrete, which 1S responsible for capacities
significantly higher than those obtained from elastic analyses.
Although emphasis was placed on behavior of the concrete, an effort was
made to model and evaluate a realistic range of soil and rock
Preceding page blank 129
stiffnesses and loading conditions. The results of the study have
provided a means of determining the ultimate thrust capacity of
concrete linings for a range of loading patterns and flexibility
ratios, representing the relative flexibility of the lining with
respect to the soil or rock medium.
From the relationships presented 1n this chapter, with the
flexibility ratio and the loading condition determined, or the initial
elastic eccentricity of the thrust in critical lining sections decided,
it 1S possible to estimate ultimate thrust levels as well as the
tensile strains and cracking that can be expected below ultimate level.
Because the concrete lining 1S often placed after ground loads have
stabilized and a nominal lining 1S capable of performing satisfactorily
regardless of the existing loading conditions 1n many ground
conditions, it is not necessary, nor is it universal practice, to carry
out structural analyses for all concrete linings. However, structural
analyses are useful in many situations, and designers do conduct such
analyses. In some cases results are obtained that can lead to
excessive reinforcement or overly thick sections, because of the
inability to adequately quantify the true nonlinear behavior of the
concrete as it approaches altimate load.
The research studies in this report have been directed toward
evaluation of these nonlinear characteristics of concrete linings. The
remainder of this chapter describes recommended approaches for
evaluating the required structural capacity of a concrete tunnel
lining, according to the following outline:
1. Determination of rock and soil loading conditions.
2. Selection of load factors to be applied to expected rock and
soil loadings 1n order to ensure that actual loads fall in an
acceptable working load range. Specific values of load factors
chosen will depend on the conservatism in the assumed loading
conditions.
3. Determination of the moment-thrust interaction diagram for
evaluating the ultimate capacity of a concrete section, and
130
'.
reduction of this capacity to account for uncertainties ~n
strength of the materials and other factors as outlined by the
ACI Code.
4. Determination of the moment-thrust path (eccentricity) at
critical sections of the lining due to application of the soil
or rock loads. Several analysis procedures based on linear
assumptions are described ~n the report and others are
available. The path is largely dependent on the flexibility
ratio and the loading pattern.
5. Determination of the ultimate thrust capacity and corresponding
load for a g~ven moment-thrust path determined in 4 (If a
nonlinear analysis was used in 4, then the ultimate thrust can
be obtained directly from the nonlinear path). The charts
6.
presented ~n this
nonlinear analyses
chapter, based on the model tests and
of the lining, permit the ultimate thrust
capacity to be determined from either known flexibility ratios
and loading conditions or from initial eccentricities
determined from the linear peam-spring analyses commonly used
by designers.
Evaluation of
determination
tensile strains at working loads and
of reinforcement requirements. The load factors
should be great enough to ensure that the actual loadings on
the lining are ~n a range where cracking and deformations are
not severe at service loads. Reinforcement requirements for
these and other conditions are discussed.
7. Applicability of the ACI Code to tunnels. Sections of the ACI
Code applicable to tunnels are noted and recommendations for
utilization of the code for tunnel linings are made.
4.2 GROUND LOADS AND BOUNDARY CONDITIONS FOR ANA.LYSIS
Ground loads depend not only on the geologic conditions at the site
and properties of the soil or rock, but also on the time and manner of
installation of the lining, presence of other support, such as initial
131
support, and assumptions regarding loading to be carried by initial and
final support, additional loads occurring over the long term due to
time-dependent effects or due to subsequent changes in loading (removal
of air pressure, build up of ground water pressure, added fill, nearby
excavation, etc.). Procedures for evaluating ground loads are
discussed in Section 1.2.
4.2.1 Soil
Sandy Soil: Linings for tunnels ~n sandy soils are usually
designed for pressures that are a function of the width of the opening.
Most analyses, model tests and field measurements show that the load
that develops ~s less than the equivalent of load due to the weight of
soil extending /2 to 2 diameters above the tunnel crown. To determine
the eccentricity, gravity loading can be used. If a beam-spring model
is used for the analysis, tangential springs should be included in the
active load region as well as where the radial springs are required.
Clay: Pressures that ultimately develop ~n soft, squeezing clays
at shallow depth are some function of overburden pressure.
Measurements show that lining pressures will increase with time (Peck,
1979), and approach overburden pressure for some clays. Peck
recommended using overburden pressure for soft clays and pressures of
p (1 + Ko )/2 for clays with high lateral stresses, where p ~s the v v
vertical soil pressure at the tunnel level.
An approximation of the load distribution and the resulting
eccentricity can be made using the elastic excavation loading analysis,
assuming reasonable values for K Time dependent and nonlinear soil o
models would more closely approximate the soil behavior.
Stiff Clays: Pressures may be as described for either sandy soil
or soft clay, depending on soil stiffness, creep and the stiffness of
the lining and its time of installatin. In most cases, tunnels at
shallow depth (less than 100 ft.) in stiff soils are capable of taking
132
full overburden pressure, using a nominal 8 to 12 ~n. thick (200 to 300
rom) concrete lining.
4.2.2 Rock
Loosening: Pressures will principally be a function of the weight
of the wedges that can loosen immediately around the opening.
Eccentricities can be determined from the loosening load analyses for
variOUS load distributions. Concentrated loads (due to small wedges)
will produce higher eccentricities, but will produce lower pressures
than will loadings form large wedges, in which eccentricities are
smaller but pressures are greater. Both tangential and radial springs
should be employed if a beam-spring model ~s used. In most rock
tunnels, eccentricities will be low enough to cause the elastic
moment-thrust path to intersect the envelope above the balance point.
Thus, linear analyses will provide a reasonably accurate estimate of
ultimate lining capacity.
Squeezing: High ground loads may develop on the lining system,
although the final lining, if installed after most of the movements
have taken place will be subject to only small pressures.
Final Concrete Linings: The loadings described for soil and rock,
although developing on the total lining system, may not fully act on
the final concrete lining installed at some later time and supported
initially by another system. The loads due to grouting, water pressure
and other time-dependent or delayed loading are discussed in more
detail ~n Section 1.2. Not only may loads on the final lining be
reduced, but eccentricities may also be smaller.
4.3 LOAD FACTORS
In ultimate strength design of concrete members, the design load to
be applied to the member is multiplied by a load factor, a quantity
133
greater than 1. The member is then designed to reach its ultimate
capacity w~en the factored load is applied. The load factor combined
with the capacity reduction factor provides a safety factor. The
safety factor is necessary to prevent excessive creep of the concrete,
possible failure of concrete due to long tenn loading exceSSIve
cracking, local spalling due to stress concentrations and to account
for uncertainty in material properties and analysis procedures. It
also IS designed to guard against overload. However, the fact that a
major part of the safety factor is provided by applying a load factor
should not be interpreted as applying the entire load factor to the
uncertainty In loads; part of the load factor is necessary to maintian
stresses and distortions in the concrete at an acceptable level when
the service ground loads are present.
Although values of load factor are specified In the ACI Code,
values are not recommended In this report because of the varying
degrees of conservatism often built into the evaluation of ground loads
In underground engineering practice. For example, in some cases, the
ground loads assumed to act on a lining may represent an upper limit or
an envelope of the possible loadings on the structure. In other cases,
all ground loads may be assumed to be applied to the concrete lining,
even though it IS known that the initial support (which carried the
initial loads) will remain in place and maintain much of its capacity,
particularly when encased in concrete. In cases such as this, where
the ground load has been conservatively estimated, the value of the
load factor to be used would be less than the value required for use
where the levels of the assumed ground loads are actually expected to
develop on the lining. If the design ground load represents an uppper
limit to the possible loadings on the lining, then it is not necessary
to include, as part of the load factor, a quantity that accounts for
the uncertainty in the loading, but it would still be necessary to have
a value of the load factor sufficient to cause the stresses and
distortions in the structure to be at an acceptable, working level when
the ground loads act on the lining.
134
For reference, load factors in the ACI Code for builings are 1.4
for dead loads, which are known with some certainty. If the capacity
reduction factor is 0.7, then the corresponding safety factor 1S 2.0
(i.e., 1.4/.7); the stress 1n the concrete will then be about 1/2
f' under the selected loads. Higher values of 1.7 are used for live c
loads that are considered to be known with less certainty. Load
factors applied in the design of permanent steel-shotcrete linings used
as both initial and final structural lining and installed close to the
face in 60 to 70-ft-wide shallow chambers excavated 1n rock are given
as examples of what has been used. A load factor of 2 was applied to
the load due to rock wedges. These wedges were observed to form, and
measured loads on the linings were 1n the range of the loading
calculated for them. The height of the rock load above the crown was
typically less than 1/2 the width of the opening. A load factor of 1.2
was applied to the full overburden load (on the order of 80 to 100 ft
of rock and soil). This loading was an upper limit, and it 1S probable
that, under the worst case, if the rock mass over the tunnel had been
allowed to loosen and collapse onto the lining, that frictional effects
would prevent the most severe loading from exceeding approximately 70%
of the overburden stress.
4.4 SECTION CAPACITY
Moment-thrust combinations obtained from the factored loads are
most easily checked by comparing them with the moment-thrust failure
envelope or interaction diagram. Use of the ACI Code (ACI 318-77)
procedure for construction of the diagram is recommended. This
procedure contains capacity reduction factors that provide additional
safety to account for uncertainty in material properties, calculation
of the resistance of the section, and the difference between concrete
strength obtained from cylinder tests and the concrete in the
structure. A capacity reduction factor of 0.7 for columns that
gradually becomes 0.9 for pure flexure is applied with a transition
near the pure flexure value.
135
The capacity reduction factors suggested are the same for linings
as for structures covered by the ACI Code because there appears to be
little reason why the uncertainties that are taken into account by
these factors should be grossly different, and the load factors should
be adjusted to provide the desired overall safety factor. Better
curing conditions in a tunnel may provide a stronger concrete, but this
advantage may be offset by the difficult conditions under which it ~s
often placed by pumping.
Shear resulting from the analysis may be compared directly with the
shear strength calculated from Section 11.3 of ACI 318-77 which takes
into account the effect of thrust. If the part of the lining for which
the shear ~s checked ~s near a corner or knee of an arch that may be
considered a support for the member, the shear should be checked at a
distance equal to the effective depth from the face of the support. If
there is no such support as ~n a circular or arch tunnel, shear should
be checked at the point of its maximum value. Shear strength of
embedded steel supports may be added to that of the concrete sections.
It would be normally unreasonable to provide shear reinforcement such
as stirrups in a tunnel lining, and therefore the thickness would
normally be adjusted to resist shear if needed. If a rock block or
wedge moves, high shear forces will be concentrated at its edges and if
the movement ~s significant, a shear failure may occur in the lining;
if the rock block is moving parallel to the discontinuity, the movement
may cause shear forces along the discontinuity to build up and the rock
block will not dislodge, but a shear failure may occur ~n the lining
that should be avoided; if the rock block movement has a component
normal to the discontinuity, the shear forces will not increase during
movement (and may reduce to zero) so the lining must resist the
movement ~n shear to prevent a drop out. Fairly concentrated loads, as
may result from small wedges of loosening rock, cause high shear at
their edges, and this condition combined with a relatively thin lining
could lead to a possible shear failure and should be checked.
136
. ,:;
4.5 MOMENT-THRUST P~TH
The relative stiffness of the lining with respect to the soil or
rock is expressed for flexure problems by the flexibility ratio,
F
where Em and E£ are the elastic moduli of the medium and lining, I£ the
moment of inertia of the lining, and R the mean radius. This ratio
largely controls the eccentricity (ratio between moment and thrust)
developed at critical sections 1n the lining. Linear analysis
procedures, both closed form and beam-spring stress analyses, are
available and have been routinely used by designers in evaluating the
thrust and moment developed 1n a lining for various loading
configurations. Such analyses, summarized in Section 1.3, have major
limitations when used to evaluate the required capacity of concrete
linings 1n conditions where the eccentricity due to the load 1S large
enough to produce significant tension in the concrete lining. Use of
linear analyses in design has in some cases led to the use of excessive
reinforcement or lining thickness to resist the computed tension. Such
an approach does not recognize the capability of an unreinforced
concrete arch to crack when in contact with the ground, yet still carry
appreciable thrust, without excessive distortion in a section having no
tensile capactity.
The model studies and nonlinear parameter studies summarized 1n
this report have provided a means for evaluating the ultimate capacity
of such linings. In these analyses the soil stiffness 1S assumed
linear, but the concrete lining is treated as a nonlinear, section,
with moments that decrease beyond certain rotations. Figure 4.1 shows
the differnce that would be obtained between the linear and nonlinear
analyses. In the nonlinear analysis, when the moment-thrust (M-T) path
approaches the interaction diagram (envelope) below the balance point,
the concrete cracks and the eccentricity decreases, resulting 1n a
137
THRUST Linear path
Nonlinear pat h -----,,,c---\
Balance point
Linear path Ie::~ ____ ~ ______ t-l
MOMENT
FIGURE 4.1 LINEAR AND NONLINEAR M-T PATHS AND PREDICTION OF FAILURE
pt' Interaction d,iagram
witbQlJt, tension
in the con~crete,
~, ! ~
M
Unstable M-T Path
"-..,
0.5 t -
MOMENT
\ \ \
Interaction diagram with tension in
the concrete 7
Nonlinear analysis path-
----.,. ... -- -- --- -~
MOMENT
FIGURE 4.2 BEHAVIOR OF AN UNREINFORCED SECTION WITH et
> O.St
138
higher value of thrust (point 2) than would be obtained in the linear
analysis when the M-T load path intersects the M-T envelope (point 1).
The section has further capacity even after the load path has reached
the envelope, and the thrust continues to ~ncrease even though the
moment capacity drops off (point 3). The results of the parameter
studies and the large scale model tests show that the concrete strains
begin to ~ncrease dramatically once the curve breaks away from the
envelope toward the thrust axis. Typically, compressive strains at the
section are ~n the range of 0.003 when the envelope is reached (Figure
3.31), but is much larger when failure actually occurs. For reinforced
concrete sections, at the point where failure occurs the thrust value
for the nonlinear case is typically 1.8 to 2.8 times the ultimate
thrust estimated from the linear analysis.
An unreinforced lining in a soft medium presents, special problems
when the eccentricity is larger than O.S t and tension in the concrete
is ignored in constructing the interaction diagram. In this case, the
M-T path will fall below the interaction envelope at the start of
loading as shown in Figure 4.2. A linear analysis would lead to the
conclusion that reinforcement ~s required to prevent failure of the
section. When tension cracking occurs, the path jumps to the cracked
interaction diagram and follows it until failure occurs in thrust.
When cracking occurs, the lining behaves as a ser~es of unbolted
segments with joints at the cracks and the failure envelope becomes the
moment-thrust path. Failure would be predicted as soon as the path
Jumps back to the interaction diagram; however, the nonlinear analysis
has shown that considerable additional strength is available as the M-T
path follows along the M-T envelope until a curvature is reached that
actually causes the section to disintegrate, moments falloff and the
ultimate thrust capacity ~s reached. Thus, if reliance ~s placed on
linear analyses for M-T load eccentricities greater than O.S/t,
reinforcement is required to produce a stable result, even though it is
well known that such a condition does not represent collapse, nor does
it necessarily produce excessive distortions and cracking.
1~
Figures 4.3-4.5 show typical results of the nonlinear parameter
studies for various loading configurations.
P applied to the lining or the ultimate thrust, u
The maXimum pressure,
T, on the initial u
lining section is plotted as a function of the flexibility ratio in
Figures 4.3 and 4.4 for a typical set of parameters. Additional
results are shown in Figures 3.30-3.38. In the excavation loading
case, the medium is assumed elastic and the lining thrusts and moments
due to immediate installation of a lining upon tunnel excavation are
determined uSing a linear finite element mesh for the continuum and
joint elements between the medium and lining. The pressure, p u'
represents the total overburden pressure when the lining reaches its
ultimate capacity. This analysis gives an actual pressure transmitted
to the lining that is approximately 70 percent of the full overburden
load. For the gravity load cases, the pressure p , is applied directly u
to the lining and then the interaction between lining and the soil or
rock medium is determined using a beam element model for the lining and
tangential springs to represent the soil or rock medium. The gravity
load is equivalent to the so-called loosening pressure resulting from
some height of rock or soil load, proportional to the width of the
opening. Tangential springs are used around the entire perimeter with
K IK 0.25, and radial springs are used only where applied load t r
causes the springs to be in compression. In Figure 4.5 all the
analysis data is shown in terms of the thrust ratio T IT versus the u 0
ratio of linear eccentricity divided by the lining thickness.
4.6 ANALYSIS OF LINING AND COMPARISON WITH ULTIMATE CAPACITY
In evaluating a tunnel lining, the plots based on the nonlinear
analyses developed in this report can be used directly to obtain the
ultimate thrust, if the flexibility ratio is known and the lining shape
(circular or arched) and loading patterns are similar to those used in
the analyses. The plot in Figure 4.5 will also permit the ultimate
nonlinear thrust capacity to be determined by first calculating the
linear eccentricity calculated from a linear analysis. The value of
140
f- .r'·
f0-
.12
• 1
-()
4-4 ----0..
0 H
• ~8 +
E--
< ;2 W
p:: ~
Cfl
Cfl .~6
W
p::
P-.
.~4
.~2
0.
o
EX
=
Ex
cav
ati
on
lo
ad
ing
O
P =
O
verp
ress
ure
lo
ad
ing
R
l80
=
R
ad
ial
un
ifo
rm
load
ing
o
ver
1
80
0
--
Rein
forc
ed
---
Un
rein
forc
ed
K
/K
=
0.2
5
t r
/ v~
--
R60
20
1
0
FL
EX
IBIL
ITY
, F 5
0
40
FIG
URE
4.
3 CO
MPA
RISO
N OF
LO
ADIN
G CO
ND
ITIO
NS
AS
FLEX
IBIL
ITY
V
ARI
ES
IN
TERM
S OF
p/f~
FOR
GRAV
ITY
LOAD
ING
AND
Kt/
Kr
OF
0.2
5
t-' ---- E--< o H E--< <t:
p::
E--<
Cfl ~
~ 0 ;j
RC
l80
=
Rad
ial
Com
pone
nt
of
vert
ical
load
ing
o
ver
18
00
R60
=
R
ad
ial
un
ifo
rm
load
o
ver
60
0
VT
=
Vert
ical
load
acro
ss
full
d
iam
ete
r
.8
.7
.6
.5
.4
--
Rein
forc
ed
---
Un
rein
fo
rced
K
/K
=
0
·25
t
r
~ ."
3
.2
• 1
0.
o 2
0
40
1 ~
FL
EX
IB I
LIT
Y,
F "3
0
FIG
URE
4.
4 CO
MPA
RISO
N OF
GR
AVIT
Y LO
ADIN
G SH
APE
S AS
FL
EXIB
ILIT
Y
VA
RIES
IN
TE
RMS
OF
Tu/
To
AND
FOR
Kt/K
r OF
0.
25
.8
.7
r .0
.S
.2
. 1
0. 0.
.5
I
1 .
o Unreinforced linings o Reinforced linings
Gravity Loading
Vertical uniform loading over full diameter
Radial uniform load over 60°
o
Radial component of vertical loading over 180 0
Radial uniform load over 180 0
o
p
2. 3. 4. 5. .5 2.5 3.5 4.5 5.5 6.5
7.
FIGURE 4.5 PREDICTION OF FAILURE OF LININGS BY THE NONLINEAR ANALYSIS AS A FUNCTION OF THE LINEAR ECCENTRICITY DIVIDED BY THICKNESS
142
the eccentricity from the linear analysis is principally a function of
flexibility ratio, load patterns and lining configuration. This
approach can be used when the lining configurations and loadings differ
from those used in the nonlinear analyses presented ~n the report. The
relationship between a given linear eccentricity and the ultimate
nonlinear thrust capacity for a lining in contact with the soil or rock
~s principally a function of the properties of the lining section and
the nonlinear characteristics of the concrete, and ~s not strongly
influenced by flexibility ratio, loading pattern, or lining
configuration. In other words, several different combinations of
flexibility ratio, loading pattern, and lining size and shape will
produce the same linear eccentricity. For all of these cases, the
ultimate nonlinear thrust capacity would be almost the same, assuming
that the concrete properties and section capacity were the same for
each case. Thus, the curve in Figure 4.5 can be used to obtain the
thrust at failure based on the computed eccentricity obtained from a
linear analysis for gravity loading.
It is also possible to use the linear analyses to determine the
eccentricity, but to modify the applied thrust levels from those that
would be predicted from this analysis, based on previous experience and
observations. For example, measured thrusts, such as those presented
by Peck for soft clays (1969) could be used to determine the thrust
level, T a
pR due to the applied pressure. The linear eccentricity
would then be determined from an analysis, such as the overburden or
excavation loading analysis. These cases, although they do not account
for the plastic and creep behavior of the clay, give reasonable values
of the linear eccentricity in the concrete section, if appropriate soil
stiffnesses have been used, and the loading pattern ~s reasonable. The
ultimate nonlinear thrust capacity would then be determined from the
given eccentricity value, using Figure 4.5. The applied thrust,
obtained from the empirical data, would be multiplied by an appropriate
load factor. This value would then be compared with the ultimate
thrust capacity represented by the interaction diagrams to ensure that
it is less than the ultimate value.
143
Caution must be exercised In such approaches, because it IS
possible to select thrust levels inappropriate to the elastic
eccentricities obtained from the analysis. In general, the higher the
thrust level, the more uniformly distributed the pressure must be, and
the lower the eccentricity. For linings in continuous contact with the
rock or soil, extremely high eccentricities can only occur for
relatively small, concentrated loads.
It is recommended that the gravity load case for soils be used with
pressures that are equivalent to a height of soil load of less than
approximately two times the tunnel diameter.
In su~~ary, to determine the maXImum allowable thrust level for a
given concrete lining, the following is recommended:
1. Below the Balance Point
a. Use a nonlinear analysis to determine where the nonlinear
M-T path intersects the reduced M-T envelope. The reduced
M-T envelope should be drawn using the
reco~~ended factors In the ACI Code to account for
concrete strength variations. Verify that the applied
thrust, T , times the load factor is less than the ultimate a
thrust, T reduced by the capacity reduction factor. u
b. Alternatively a linear analysis may be used to determine
the linear eccentricity of the thrust for the gIven
flexibility and loading conditions, and then the chart of
Figure 4.5 may be used to obtain the T (nonlinear); this u
value IS then compared with the applied T times an
appropriate load factor.
2. Above the Balance Point
a
The same procedures outlined In (1), above, can be utilized in
evaluating the capacity of a lining where the linear moment-thrust path
intersects the envelope above the balance point. However, above the
balance point, the ultimate thrusts calculated from the linear analysis
will be closer to that obtained with a nonlinear analysis than it is
below the balance point, so the linear ana~ysis can be used directly
without being excessively conservative.
144
..
. .... ~
For most linings in rock, and for linings in many of the stiffer
soils, the moment-thrust path will approach the envelope above the
balance point so the linear analysis can be used.
4.7 EVALUATION OF STRAINS AND CRACKING
The actual thrust levels determined from the assumed loads (without
load factors) should fall low enough along the moment-thrust path that
distortions and tensile strains do not cause excessive damage.
Concrete will creep in tension as well as compression so if the
loads are applied slowly, larger tensile strains can occur without a
crack forming. Therefore, for a final lining placed Ln a presupported
opening where considerable time will be required for the loads to reach
the final lining it is reasonable to double the allowable cracking
strain when a cracking analysis LS performed. A reasonable value of
allowable tension strain to avoid cracking from this criteria would
then be on the orqer of 0.0003 in./in. Figures 4.6 and 4.7 from the
nonlinear analyses show the level of tensile strains developed Ln both
unreinforced and reinforced concrete sections for typical cases of
gravity loading Ln a soft media. Maximum tensile strains of 0.001
in./in., at the center of a 90 degree arc of the lining and dropping to
zero at the edges of the arc would produce a total width of a single
crack of approximately 0.1 in. (two.5 rom) for a 10-ft-radius (3.0 m)
lining. If longitudinal cracks were spaced 12 in. (300 rom) on center,
the maxLmum crack width would be close to the 0.01 in. crack level,
recognized as a working limit in some concrete pipe design. The 0.0003
strain level, assumed to be the initiation of cracks, would produce
crack widths exceeding the 0.01 Ln. (0.25 rom) level only for crack
spacings greater than approximately 3 ft. Concentration of the
deformation could occur in a few cracks for a non-reinforced lining,
causing an increase Ln the width of the individual cracks. This
condition would be more likely to occur where concrete lining thickness
LS variable around the perimeter, perhaps due to overbreak, and where
145
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lining-to-rock or lining-to-soil contact ~s poor.
temperature effects would influence cracking, also.
pronounced cracking occurs due to shrinkage, often
circumferential direction.
Shrinkage and
Usually the most
forming ~n the
To ensure that cracking at the 0.001 in./in. tensile strain level
~s not excessive, a light reinforcement may be needed to spread cracks.
At the 0.0003 in./in. strain level, crack widths should not be
excessive due to lining distortion, even for non-reinforced sections.
The model tests discussed in Section 3.1 shows that reinforcement
in a lining does little for strength in the amounts usually used for
underground supports. However, reinforcement ~s effective ~n
distributing cracks and therefore keeping each crack from becoming
large. When no reinforcement was used in the model tests, only one
crack would form in each high moment region, but if reinforcement was
present, the cracks would occur 4 to 6 in. (102 to 152 mm) apart and
would be much finer. On the other hand the tests also showed that
cracks first appeared at loads 50 percent or greater than the ultimate
load. This percentage would be larger if the loading was appplied very
slowly. Therefore, flexure cracks are not common at working loads and
flexure reinforcement is not recommended to control them unless there
is a definite need shown by calculations. If cracking appears to be a
problem, a minimum amount of reinforcement of from 0.25 to 0.50 percent
should be used on the tension side of the highly stressed region, but
is not necessary throughout the lining.
The ma~n reason proposed for limiting flexureal cracking ~n a
lining ~s to prevent leakage. However, when a crack occurs a
substantial area of the section is still in compression, and the stress
becomes larger as the area is reduced. This compressed concrete should
remain an effective barrier to water penetration. On the other hand,
circumferential cracks due to shrinkage penetrate through the section
and become a more likely path for leakage. The longitudinal flexural
cracks due to ground loads are probably much less a problem insofar as
147
leakage ~s concerned than shrinkage cracks, joints or casting flaws
that might occur.
4.8 APPLICATION OF ACI CODE
Many provisions of ACl 318-77 do not apply to underground
construction, others require modification, and a few may be used
without change; Chapter 4 "Concrete Quality" and Chapter 5 "Mixing and
Placing Concrete" describe standard requirements for preparing and
placing concrete and in general, apply to underground work. Quality
concrete without flaws is required for strength and to prevent leakage.
The transit time without agitation is sometimes longer ~n tunnel work
than above ground and therefore larger slumps are required at the mix
plant to assure workability at placement. At placement the Code
requ~res that "Concrete shall be deposited as nearly as practicable in
its final position to avoid segregation due to rehandling or flowing."
since the concrete for linings ~s normally pumped into the cavity
behind the linings at the crown or through windows at the springlines,
a considerable amount of flowing ~s required to get concrete
distributed around the forms; techniques have been developed ~n the
tunneling industry to reduce segregation ~n this case and a
satisfactory placement ~s attained even when there ~s considerable
flowing of the concrete. Additives such as the var~ous
superplasticizers have helped greatly ~n this regard by improving
strength while also improving workability without segregation.
Chapter 6 "Formwork, Embedded Pipes, and Construction Joints" has
two sections that apply to cast linings. Section 6.2 "Removal of Forms
and Shores" is adequate ~n principle, because it states that forms may
not be removed until the structure and any remaining shoring can
support its weight and any loads that may be on it. Since the ground
~s stable when the final lining is placed, either due to the use of
initial supports or the integrity of the ground, then the only load
that should occur ~s the weight of the concrete lining itself. With
148
this self weight applied to
maximum concrete stresses,
the lining, an analysis
with allowances ~n the
to determine
analysis for
interaction with the ground, may be performed. Once the max~mum
stresses are known, the acceptable compressive stress to be reached by
the concrete must be selected.
At an early age, concrete tends to creep more rapidly under load
than it does when fully cured. Creep may allow deformation of the
lining and separation from the ground at the crown if the forms are
removed prematurely. Normally deformations will be negligible if the
concrete strength has reached four times the maximum compressive stress
~n the concrete. With the required compressive strength of the
concrete determined, it is necessary to monitor the gain of strength in
the lining with time, to determine when the required strength has been
reached. This may be done with field cured cylinders or by other
means. However, the. rate of strength gain ~s influenced by many things
such as water content, temperature, cement formulation and fineness of
grind. Therefore, the time at which a given strength is reached may
change suddenly without an obvious cause, and close monitoring is
required. This approach to removal of forms allows greater latitude to
the contractor in his casting operation and provides the opportunity to
~ncrease the casting rate by increasing the strength gain of the
concrete through the use of additives or more rapid set cement types.
Construction joints are necessary to separate castings when the
operation cannot be continuous. Specially prepared vertical joints are
often specified and are rather expensive because of the labor and time
required for their placement. It is proposed that a sloping joint ~n
which the concrete is allowed to assume its own angle of repose will be
adequate, and ~s no more likely to leak than a vertical joint without a
special water stop. If separation of the joint occurs due to shrinkage
of the concrete, the opening is actually less for a sloping joint than
for a vertical one. When this type of joint ~s used, the surfaces
should be prepared as described in Section 6.4 of ACI 318-77 before the
next pumping cycle.
149
Chapter 7 "Details of Reinforcement" contains provisions for
protection of reinforcement and to assure that it is effective when
used. Therefore, the sections on bending and surface conditions apply.
The tolerances on placement may be relaxed because of the difficulty in
placing bars, but the specified cover should be increased to compensate
for the changes 1n tolerance and greater tendency to rust. Concrete
cover over reinforcement is specified as 3 in. (76 mm) for concrete
cast against the ground and appears appropriate for cast in place
tunnel construction. Reinforcement on the inside face should have at
least 2 1n. (50 mm) of cover as required by the ACI Code for
construction exposed to the weather. Though the tunnel surface is not
actually exposed to the outside weather it may be exposed to frequent
wetting due to condensation on the surface that will eventually reach
the inside bars. In many old tunnels much damage has occurred due to
spalling of concrete due to expansion of corroding bars that exposed
the bars and accelerated the process. In precast construction where
the concrete is placed with greater control and is of higher quality,
it may be possible to further decrease the cover without causing long
term corrosion.
At the crown where inside reinforcement is in tension, there is an
inward force on the concrete around the bars when moment occurs and the
region flattens, or the radius of curvature is increased. The cover in
this region should be checked to assure that the concrete tension force
around the bars is adequate to hold them in against the radial force
that results when tension tries to straighten the bar. If the diagonal
tension in the concrete along the bar 1S inadequate, it may be
necessary to 1ncrease the cover in this region. The crown region of
the lining 1S sometimes flattened on the inside to allow additional
room for the pumping slick line to be inserted above the form. This
practice eliminates the problem discussed above and also increases the
bending strength of the lining in the crown region.
The sections on lateral confinement of bars in compression by ties
and stirrups are not appplicable. Horizontal confinement 1n the
longitudinal direction of the tunnel is not necessary because the
150
lining is continuous in this direction. Confinement outward is
provided by the rock or soil on the outside of the lining. Also the
inside bars are in tension in the crown region, and on the sides where
they are in compression, the original curvature is outward so their
tendency to buckle is toward the inside of the member (outside of the
tunnel) where confinement is adequate.
Chapter 9 "Strength and Serviceability Requirements" describes the
load factors, capacity reduction factors and control of deflection and
cracking that were discussed previously. Various considerations in the
control of cracking are discussed by ACI Committee 224 (ACI-224, 1972)
where means are provided for estimating crack size and distribution due
to shrinkage and flexure. The formulas presented there for maX1mum
crack width due to flexure are written in terms of the tension
reinforcement stress and in this form can be applied to members with
axial load, as well as flexure. The axial compreSS1on results in a
uniform stress that reduces the flexural cracking by reducing the
tension stress in both the concrete and tension reinforcement. Use of
these formulas will result in excessive additional reinforcement when
used with 3 1n. (76 mm) of cover, however, because they were devised
for slabs and framed walls with less than 2 1n. (51 mm) of cover.
Recognizing this problem, CRSI Bulletin PSI-7804A recommends using a
maximum value of 2 1n. (51 mm) for cover in crack calculations when the
actual cover is greater than 2.0 in. (51 mm). Also crack width
normally need only to be checked on the outside face for most
transportation tunnels in non-public places.
Chapter 10 "Flexure and Axial Loads" concerns the calculation of
strength and this topic has been discussed. The methods for
calculating strength presented there have been recommended. The limit
on minimum flexural reinforcement in this chapter is to assure that the
ultimate flexural strength is larger than the cracking strength to
avoid failure upon cracking. This provision does not apply to linings
because failure would not occur at cracking since the force within the
lining would be redistributed as a result of the high degree of
redundancy and confinement of the ground.
151
Chapter 11 "Shear and Torsion" can be appl ied to calculate the
shear strength of linings as described in Section 11.3 where provisions
are included for considering the axial force effects on shear.
Chapter 12 "Development and Splices of Reinforcement" should be
applied to lining design in order to assure that needed reinforcement
will be effective. Development of bar forces are primarily a function
of the bar geometry and concrete strength, and therefore does not
change for underground structures.
4.9 SUMMARY
The recommendations given ~n the preceding sections were developed
as a result of analyses and model testing of the behavior of concrete
tunnel linings. The research addressed problem areas in current design
practice, and the results have provided insight into the areas of
uncertainty that have led designers to overconservatism ~n tunnel
lining design.
The recommended procedures provide sufficient latitude for
designers to exercise judgment gained through experience and allow the
flexibility required by site-specific conditions. Details of the
suggested approach are based upon procedures that have been accepted
for years ~n the design of above-ground structures, with appropriate
modification to capitalize on the benefits of ground/structure
interaction.
152
,. ; ~ ~
ACI Committee 224, Structures," Journal No. 12, pp. 717-757.
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